THE INFLUENCE ON. A FINITE GROUP OF THE COFACTORS AND SUBCOFACTORS OF ITS SUBGROUPS Thesis for the Degree of Ph. D. MICHIGAN: STATE UNIVERSITY LARRY RAY NYHOFF 1969 _—... :sk; ‘2“;‘4- k“ ‘. v (’0’ hi. lCl 3.1.. i ex ea Ufllvcfbity I; “A -. hate w cg. mew-n. This is to certify that the thesis entitled THE INFLUENCE ON A FINITE GROUP OF THE COFACTORS AND SUBCOFACTORS OF ITS SUBGROUPS presented by Larry Ray Nyhoff has been accepted towards fulfillment of the requirements for Lu..— degree in We: Major professor Date July 231 1969 0-169 IIN‘DING BY m: a sour lam: mm m. LIBRARY IINDEIS 4 ABSTRACT THE INFLUENCE ON A FINITE GROUP OF THE COFACTORS AND SUBCOFACTORS OF ITS SUBGROUPS BY Larry Ray Nyhoff There are a number of theorems of the form: If every proper subgroup of the finite group G has property X, then G has property Y. Examples are the classic results of Schmidt, Iwasawa, Ito, and Huppert. Such results have been extended by imposing condition X on only one maximal subgroup of G (Deskins, Huppert, Thompson) or on a certain class of subgroups (Rose). The major goal here is to extend results of this type by imposing condition X on only the "worst" parts of the "bad" subgroups of G (from the vieWpoint of normality), namely, the cofactors or subcofactors of the self-normalizing or abnormal sub- groups of G. In some cases, x is also imposed on the "good" sub- groups, those whichare normal or close to being normal in G. In the last chapter, the influence on G of X-outer cofactors of subgroups is examined. Throughout, C denotes a finite group. For a subgroup H of G, the cofactor and subcofactor of H are cof H = H/cor H and scofGH 8 H/scorGH G G respectively, where corGH is the core of H in G, and scorGH = the sub- normal core of H is the largest G—subnormal subgroup of H. For X - nilpotency, we have: G/F(G) is nilpotent if-f cofGS is nilpotent for all maximal subgroups S of G, where F(G) is the Fitting subgroup of G. Also, if yn(G) is the (n+l)st term of the descending Larry Ray Nyhoff central series of G, the following are equivalent: (a) G/F(G) has class g n. (b) cofGS has class a n for all maximal subgroups S of G. (c) yn(G) is nilpotent. (d) ynO-I) <1 4 G for all (abnormal) H s G. Under the hypothesis that G is solvable, one can replace Yn by the (n+l)st derived subgroup and "has class é n" by "has derived length a n" in (a)-(d) above. The resulting statements are equivalent. In the preceding results, G need not have a normal Sylow sub- group. If, however, each K‘a G also is nilpotent, but G itself is not, then for some prime pr where ‘Gl = 151 p181, we obtain, among other results: (1) |G:F(G)| 8 pr so that G has normal pi-Sylow subgroups P1 for each i f r, and each Pi c G'. (2) ‘H/scorGHI - 1 or pr for each a H‘s G. (3) For each abelian P i f r, G has pi i pi-complements, is CG(P1) ‘ F(G), and pia1 divides the number of pr-Sylow subgroups of G. (4) If all the Pi’ i f r, are abelian, then G has IG‘lprar pr-Sylow subgroups. Under the preceding hypotheses, ‘n(G)| can be arbitrarily large. Defining Hi< G to be nearly normal in G if ‘H/corGH‘ . 1 or a prime, however, we have that if G is nonnilpotent but has all nearly normal maximal subgroups nilpotent as well as the cofactors of maximal sub- groups, then ‘n(G)‘ ' 2, all proper subgroups of G are nilpotent, and thus the Schmidt-Iwasawa conclusions hold. For X 3 p-nilpotency, we have: If (a) scof H is p-nilpotent for G each self-normalizing H‘s G, or if (b) scofGH is p-nilpotent for each abnormal H‘g G and either p is odd or the p-Sylows of G are abelian, _then, in each case, G has a normal p-subgroup P with G/P p-nilpotent. If in addition to (a) or (b), each K‘a G is p-nilpotent while G itself is not, then, among other things: (1) G has a normal p-Sylow subgroup P'with P’: G'. (2) If P is abelian, then qG(P) = Fp(G), the Larry Ray Nyhoff largest normal p-nilpotent subgroup of G. (3) For G solvable, ‘G:Fp(G)‘ is equal to a prime * p; and if also P is abelian, G has exactly pa distinct p-complements, where ‘Gl . pam'with (p,m) = 1. If (a) or (b) as above holds and each proper somewhat normal subgroup of G is p-nilpotent, but G is not, where H.< G is somewhat normal in G if H/corGH is cyclic of prime-power order, then ‘G‘ = paqb for some prime q * p, and the Ito-Schmidt-Iwasawa conclusions hold. Defining G to be (p:q)-nilpotent if (i) G is p-nilpotent, and (ii) G is q-nilpotent with q||Gl in case p I ‘G‘ and |G| >»l, we obtain: If each x‘fi G is (p:qK)-nilpotent and cofGH is (p:qH)-nilpotent for each H“ G, then G is solvable and has a normal Sylow subgroup (in addition to (1)-(3) above holding in case G is not p-nilpotent). For X ' supersolvable or Sylow-towered, cofGS supersolvable for all maximal subgroups S of G does not imply that G is solvable. But for a fixed ordering o of a set of primes containing n(G), we have G solvable with G/F(G) a-Sylow-towered if either (a) scof H is o-Sylow- G towered for each self-normalizing H'$ G, or (b) scof H is o-Sylow- G towered for each abnormal H‘s G and the 2-Sylows of G are abelian. If (a) or (b) holds with "supersolvable" replacing "o-Sylow- towered," or if (c) scofGH is supersolvable for each abnormal H‘i G and the abnormal maximal subgroups of C have prime-power index, then, in each case, G is solvable with G/F(G) supersolvable and Fitting length of G' "f(G') § 2, f(G) § 3. These are the best possible bounds on fm') and f(c). In the last chapter, the outer cofactors of a subgroup H as a kind of dual to cofGH are considered. These are of the form C/corG(C 0 H) where C ¢ H and L c H for each proper G-normal subgroup L of C. We term this a normal, self-normalizing, or abnormal outer cofactor of H Larry Ray Nyhoff according as C is normal, self-normalizing, or abnormal in C. General results for nonnormal outer cofactors, from which corollaries parallel to the preceding results follow, are: Given a subgroup-inherited pro- perty e which is invariant under homomorphisms. If (a) G has a 9-maxi- mal subgroup whose self-normalizing (abnormal) outer cofactors are 9- groups, or if (b) the self-normalizing (abnormal) outer cofactors of each abnormal maximal subgroup of G are e-groups, then, in either case, cofGH is a e-group for all self-normalizing (abnormal) H‘s G- The following are some of the properties of the normal outer cofactors. For a maximal subgroup S of G, the normal outer cofactors of S are isomorphic; the order of any one is called the normal index of S. The normal outer cofactors of S are p-solvable (solvable) if-f the normal index of S is a power of p or is prime to p (is a power of a prime); in the solvable case, the normal index and the index of S are equal. The influence on G of normal outer cofactors is described by: (a) G is p-solvable if-f (b) G has a p-solvable maximal subgroup having p-solvable normal cuter cofactors if—f (c) the normal outer cofactors of each (abnormal) maximal subgroup of G are p-solvable if-f (d) the normal index of each (abnormal) maximal subgroup of G is a power of p or is prime to p. If "p-solvable" is replaced by "solvable" and "is a power of a prime" replaces "is a power of p or is prime to p," the resulting statements are equivalent to (e) the normal index and the index of each (abnormal) maximal subgroup of G are equal. Finally, the intersection of all the maximal subgroups of G with normal index divisible by both p and some prime f p coincides with the intersection of all abnormal maximal subgroups having this prOperty, Larry Ray Nyhoff and is equal to the largest normal p-solvable subgroup of G. Replac- ing "p-solvable" by "solvable" and "divisible by both p and a prime # p" by "divisible by two distinct primes" yields an immediate corollary. THE INFLUENCE ON A FINITE GROUP OF THE COFACTORS AND SUBCOFACTORS OF ITS SUBGROUPS BY Larry Ray Nyhoff A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1969 TO MY WIFE AND CHILDREN ii ACKNOWLEDGMENTS I wish to express my sincere gratitude to Professors Joseph E. Adney and W. Eugene Deskins, my major professors, for the many suggestions, the guidance, and the understanding ear they have afforded during the preparation of this work, to my wife and children who have patiently and understandingly endured the hardships of the past months, and to the Great Mathematician, God Almighty, for sparing me to see this culmination of efforts. iii TABLE OF CONTENTS Page CHAPTER 1: INTRODUCTION; PRELIMINARY RESULTS 1 CHAPTER 2: THE INFLUENCE ON A GROUP OF THE COFACTORS AND SUBOOFACTORS OF ITS SUBGROUPS 2.1 BaSic Results O..0.0......O.COCOOIOOOOOOOOOOOOOO00.... 8 2.2 Nilpotent Cofactors or Subcofactors .................. 10 2.3 p-nilpotent Cofactors or Subcofactors ................ 28 2.4 Sylow-towered and Supersolvable Cofactors or SUbCOfaCtors OOOOOOOOOOOOOOOCOOOCOOOO00......0..0.. 47 CHAPTER 3: THE INFLUENCE ON A GROUP’OF THE OUTER COFACTORS OF ITS SUBGROUPS 3.1 Introduction; Definitions and Basic Properties ....... 64 3.2 Influence on a Group of the Nonnormal Outer Cofactors of Subgroups ......................... 66 3.3 Influence on a Group of the Normal Outer Cofactors of Subgroups ......................... 72 3.4 Addendum .... ......... ...... .......................... 86 BIBLImRAPIIY 0.0.0.0....0.00.00.00.00.000000000000000 ...... 0.0 88 iv CHAPTER ONE INTRODUCTION; PRELIMINARY RESULTS There are a number of known theorems of the type: For a finite group G, if every proper subgroup of G has property P, then G has pro- perty Q. For example, Schmidt [18] and Iwasawa [14] have shown that if every proper subgroup of a finite group G is nilpotent, then G is solv- able. More precisely, if G itself is not nilpotent, then ‘G‘ = paqb for distinct primes p and q; G has a normal p-Sylow subgroup P with 9(P) c Z(G), and thus P has class i 2; exp(P) = p or exp(P) a 4 accord- ing as p is odd or p = 2; each q-Sylow subgroup Q of G is cyclic with ¢(Q) C Z(G). Euppert [12] and Doerk [6] have obtained corresponding results for the case where the proper subgroups of G are supersolvable. Results are also known for the cases where the proper subgroups of G are p-nilpotent, or abelian, or o-Sylow-towered for some fixed ordering a of a set 2 of primes containing the prime divisors of 'G‘. Extensions of such results have been obtained by imposing the conditions not on the totality of proper subgroups, but only on certain subgroups. Thus there are a number of theorems in which the conditions are imposed on only one maximal subgroup of G. For example, Deskins has shown in [4] that the finite group G is solvable if it possesses a nilpotent maximal subgroup having Sylow subgroups of class a 2. There are also a number of results in which the conditions are imposed on the proper subgroups of a certain kind. Examples of this are the following two results established by Rose in [17]: (1) If all the proper ab- normal subgroups of the finite group G are nilpotent, then G is solvable; in fact, G has a normal Sylow subgroup P such that G/P is nilpotent. 1 (2) If all the preper self-normalizing subgroups of the finite group G are supersolvable, then G is solvable. In the following chapters, the major effort is directed at extending theorems of the type described. We shall, like Rose, con- sider the influence on a finite group G of conditions imposed on those prOper subgroups of G which, from the viewpoint of normality, are the "bad" subgroups, namely, the self-normalizing subgroups of G, or the abnormal subgroups of G. However, we will not require that the con- ditions be satisfied by these subgroups themselves, but only by their "worst" parts (from the viewpoint of normality or subnormality), that is, their cofactors, or subcofactors (or, as in Chapter 3, their outer cofactors). In all the cases we consider, this will be enough to guar- antee the solvability or p-solvability of G, and in several cases we can say more. To obtain still more information about the structure of G, we shall on occasion impose the conditions on the "good" sub- groups of.G also, that is, the normal subgroups, or the subgroups of G which are rather close to being normal in the sense that their co- factors are quite small. In the last chapter, we define the outer co- factors of subgroups of G, as a kind of dual to the cofactors, and investigate their influence on the group G. To begin, therefore, we make the following definitions. Definition 1.1: For a proper subgroup H of the finite group G, we define: (i) the £252.2£_H in G, cor H, as -—- c corGH = fl Hx XGG = the largest G-normal subgroup of H; (ii) the subnormal core of __ in Q, scorGH, as scor H = the largest G-subnormal subgroup of H G I=. Note: It is a well-known property of subnormal subgroups (see, for example, Scott [19], 15.2.4) that if L1 and L2 are subnormal sub- groups of a finite group G, then < L1 , L2) is also subnormal in G. Thus the subnormal core of a subgroup H of G is well-defined. Le 1.1: For a prOper subgroup H of the finite group G, scorGH is normal in H. Proof: From the definition, L = scor H is subnormal in G, say G 1.4 N110 ... d Nr = C. Now, let x be any element of H; then, clearly, Lx also is subnormal in G. From the definition of L = scorGH it follows that L3 = L. Thus (scorGH)x = scorGH for each x E H so that scorGH d H. U The preceding lemma makes possible the second part of the following definition. Definition 1.2: For a proper subgroup H of the finite group G, we define: .__._.__._ GH, as. CH; (ii) the subcofactor 22.3.12.§: scof cofGH = H/cor GH, as scofGH H/scorGH. Since we shall be dealing only with finite groups, we assume at the outset that all groups considered here are finite. It might be mentioned, however, that the preceding definitions of the core and the cofactor of a subgroup of a given group G are still legitimate in the case that G is an infinite group. Also, one can show (see, for example, Scott [19], 3.3.5) that if G possesses a prOper subgroup K of finite index, then G/cor K is a finite group. Thus, the results obtained do G give some information about such infinite groups. For suppose that some group-theoretic property 9, which is preserved under homomor- phisms, is required of the cofactors (and/or cores) of subgroups of G, and that K is a proper subgroup of G of finite index. Then the cofactors (and/or cores) of proper subgroups of G/cofGK also are 9- groups (since (i) and (iii) of Lemma 1.4 hold for any group G). Thus, for example, supposing that the cofactors of all maximal subgroups of G are nilpotent as are the prOper normal subgroups of G and that G has a proper subgroup K of finite index, we have that G is solvable and the conclusions of Theorem 2.11 hold for G/corGK. Leg g‘ng: Let 9 be a homomorphismrinvariant property, that is, homomorphic images of e-groups are e-groups; and let H be a proper subgroup of the finite group G. If cofGH is a e-group, then scofGH also is a e-group. ‘ggggg: This is immediate; for it follows from the definitions that corGH C scorGH. Thus, scofGH = H/scorGH is a homomorphic image of the e-group cofGH = H/corGH, and hence is also a e-group. D It follows from the preceding lemma that any theorem*which gives information about G resulting from conditions imposed on the subcofactors of subgroups of G will automatically be true if these conditions are satisfied by the cofactors of these subgroups (provided, of course, that these conditions are homomorphism-invariant properties). Conse- quently, wherever possible, we will impose conditions on only the sub- cofactors of subgroups as opposed to their cofactors. The subnormal core of a subgroup H of a finite group G is in general not equal to the core of H. One need only take H and C so that H did C but Hid G to see this. For example, if G = A = the alternating 4 group of degree 4, and H is a subgroup of order 2, then corGH = < l>, but scor H = H. For maximal subgroups, however, the core and subnormal G core must always coincide. Lemma 1.3: If S is a maximal subgroup of the finite group G, then corGS I scorGS. Proof: From the definition, scorGS is subnormal in G, say scorGS ‘3 N1 $ N2 fl fl Nr 8 G; and no subgroup H of S properly con- taining scorGS can be subnormal in G so that, in particular,‘H1 ¢ S since N114<< G. From Lemma 1.1 we have scorcsia 8; also, scorcsid‘Nl. Therefore, scorGS d = G, which implies that scorGS C corGS. Since the reverse inclusion is immediate from the definitions, this establishes the desired equality. fl Because many of our results will involve induction arguments, we must examine for a given group G the relationship between the core and the subnormal core of a subgroup of a homomorphic image of G and those of the corresponding subgroup of G. The following basic lemma does precisely this. ‘L29 5 1:4: Let H and K be proper subgroups of the finite group G with K«d G and K C H. Then: (i) cor G/K (ii) scorG/KOI/K) . scorGH/ K; (H/K) = corGH/ K; (iii) cofG/K(H/K) E cofGH; (iv) scofG/K(H/K) : scofGH. Proof: (i) Let L/K ' cor (ll/K). Since L <1 G and L C H, we G/K have L c: cor H; thus cor (H/K) c corGH/ K. Conversely, since K <1 G c G/K and K c H, we have K s: cor H; and since cor H <1 G and cor G G G (H/K) = corGH/ K. H:H,we have corGH/K C corG/K (ii) is proved in a similar way. (H/K) . Therefore, corG/K (iii) From the definition of cofactor, part (i), and the Third Isomorphism Theorem, we have g H/K : H/K = cofG/KCH/K) COIG/Km/K) W '* H/corGH cofGH. (iv) follows in a similar manner. H The three results that follow illustrate the close connection between the cores and subnormal cores of subgroups of a finite group G and the normal structure of G. Leggg l;§: Let G be a finite group. Then: (i) cor H i < l> for all proper subgroups < l> i‘ H i G if and C only if every minimal subgroup of G is normal in G. (ii) scorGH 3‘ < l> for all preper subgroups < l> f H $ G if and only if every minimal subgroup of G is subnormal in G. M: (i) Suppose first they every nontrivial proper subgroup of G has nontrivial core, and let M 3‘ < l> be any minimal subgroup of C. By hypothesis, corGM # < 1); thus, by the minimality of M, we have M = corGM is a normal subgroup of G. Conversely, suppose that every minimal subgroup of G is normal in G, and let H be any nontrivial prOper subgroup of G. Taking it E H of prime order, we have that since is a minimal subgroup of C, it is, by hypothesis, normal in G. Thus, < l> 1‘ C cor H 80 that G corGH is nontrivial. (ii) is proved in a similar manner. I] Theorem 1.6: For a finite group G, corGH is nontrivial and is Hall in H for all proper nontrivial subgroups H of G if and only if all subgroups of G are normal in G, that is, G is a Dedekind group. nggf: Suppose first that every prOper nontrivial subgroup H of G has a nontrivial core which is a Hall subgroup of H, and let K be any subgroup of G. Since the trivial subgroups 'and G are normal a r in G, we may assume that # x'? G. Let ‘K‘ = H p,1 where the pi i=1 are distinct primes dividing lK‘; and for each i = 1,... , r, let xi 6 K have order pi' Then, from Lemma 1.5, <1: >‘d G for each i, and 1 thus L =... 4 G, if x €‘< H , Hx > for every x E G, or equivalently, if every sub- group of G containing H is self-normalizing in G and H is not contained in two distinct conjugate subgroups of G. The following remarks are immediate consequences of this definition and familiar results from Sylow theory: (1) IfHXGandHcKEG, thenKXG. (2) A maximal subgroup S of G is abnormal in G if-f S is self-normalizing in G if-f S is nonnormal in G. (3) For every Sylow subgroup P of G, NG(P) >0 G. As a first result describing the structure forced upon a finite group by nilpotent cofactors or subcofactors, we take 9 - nilpotent in Corollary 2.2-(i). Since nilpotence is clearly a strictly homomorphism- invariant property, we obtain the following result. Theorem ggi: For G a finite group, G/F(G) is nilpotent if and only if cofGS I S/corGS is nilpotent for all maximal subgroups S of G. In slightly different terminology, this theorem states that: F(G) a 2 if-f cof S is nilpotent for all maximal subgroups G S of G, where [(G) denotes the Fitting length of G and is defined as follows. Definition 2.5: The ascending Fitting series of the group G, < 1> - F0(G) c F1(G) : : Fi(G) c F (G) c i+1 is defined by FO(G) =.< l >, and F (G)/Fi(G) ' F(G/Fi(G)), 1+1 the Fitting subgroup of G/Fi(G). The Fitting length of a 12 finite solvable group G, §(G), is the least integer n for which Fn(G) ' C. One would hope to be able to say more if the cofactors of the maximal subgroups are all required to be nilpotent of the same class. Before stating the result, however, we first recall the definition of the class of a nilpotent group. Definition 2.6: The descending central series of a group G, G ' vO(G) 2 y1(G) :2 :2 yi(G) :2 yi+1(G) 2 is defined by v0(G) = G and yi+1(G) -= [vi(G) , G] - the sub- group generated by all commutators [x,y] = x-ly-lxy, where x E yi(G) and y E G. The 21223 of a finite nilpotent group G, cl(G), is the least integer n for which yn(G) =.< 1 >3 The following remarks are immediate consequences of this definition: (1) If ¢ is a homomorphism of G onto G, then yidi) ¢(vi(G)) for each i. (2) yi(G/Y1(G)) =‘< l > for each i. £22 a ggg: Let Fn denote the property "nilpotent of class é n." Then: (i) For any group G, [G,Fn] = yn(G), where [G,Fn] is the Fn-commutator subgroup of G. (ii) Fn is a strictly homomorphism-invariant property. 2593;: (i) [emu] -= n{1< 4 clc/x is a I‘m-group] '= n{1< <1 len(G/K) = < 1 >} = ntx <1 G|Yn(G)K/ K = < 1 >} (by remark (1) above) = MK <1 Chum) ‘3 Kl = vn (G) 13 (ii) Let G be any I‘D-group, that is, yn(G) = < 1 >, and let H be a subgroup of C. It is clear from the definition that Vi (H) C Yi (G) for each 1; thus, Ynm) I < l > also so that H is a Fn-group. There- fore, Fn is a subgroup-inherited property. From remark (1) above, it follows that Fn is homomorphism-invariant. And from remark (2) above and part (i), we have yn(G/[G,I‘n]) I yn(G/'Yn(G)) I < l > so that G/[G,I"n] is a I‘n-group. Consequently, I" is a strictly homo- morphism-invariant property. I] Theorem 2.7: For a finite group G, the following are equivalent: (i) G/F(G) is nilpotent of class .5. n. I (ii) cofGS I S/corGS is nilpotent of class 5 n for all maximal subgroups S of G. (iii) yn(G) is nilpotent. (iv) Yam) 4 4 G for all subgroups H of G. (v) YnCH) 4 4 G for all proper abnormal subgroups H of G. 2592;: The equivalence of (i) and (ii) is an inmediate con- sequence of Lama 2.6 and Corollary 2.2- (i); and (ii) _. (iii) follows from Theorem 2.1- (i) and Lemma 2.6. (iii) .. (iv). If H is any subgroup of G, then yn(H) s: yn(G). Since yn(G) is nilpotent, Ynm) is subnormal in yn(G) which is normal in G; hence, yn(H) 4 4 G. (iv) ... (v) is trivially true. (v) .. (ii): Let S be any maximal subgroup of G. If S 4 G, then cofGS I < l > obviously has class é. n. Thus suppose S 4 G, and hence S >4 G. Then, since yn(S) 4 4 G, we have, using Leanna 1.4, that vn(S) c scorGS I corGS. Now, by the remark (2) above, yn(S/vn(8)) I < 1) so that S/yn(S) is a I‘n-group, and hence, by Lemma 2.6-(ii), so 14 also is its homomorphic image S/corGS. Therefore, cofGS = S/corGS is nilpotent of class a n. n We obtain a similar result using the property An: solvable of derived length a n. Here we mean, as usual, by the derived length of a finite solvable group G, the least integer n for which the term.G(n) of the derived series of G is trivial; the derived series of G, G ,3 cm) (1) (1) :2G 2...:G (1+1):2. 2 G is defined by 6(0) = G and G<1+1> = (C(i))', the derived subgroup of C(i). As before, the following remarks are immediate consequences: (1) (i) (1) If ¢ is a homomorphism of G onto G, then (G) I ¢(G ) for each i. (2) (G/G(i))(i) = < 1> for each 1. Lemma 2.8: Let An denote the property "solvable of derived length E n." Then: (n), where [G,An] is the (i) For any group G, [G,An] I G An-commutator subgroup of G. (ii) An is a strictly homomorphism-invariant property. 22229 (1) Lean] = n{1< 4 G‘G/K is a An-group] = MK 4 cl (G/K) (n) < 1>} = O[K 4 G|G(n)l(/ K < 1)] (by remark (1) above) I n{1< 4 Glam) : K} . G(n) (ii) Clearly An is subgroup-inherited; and by remark (1) above, it is homomorphism-invariant. From part (i) and remark (2) above, we ) (n) (n) (n) have (cs/[cash] - (G/G ) group. Thus, An is a strictly homomorphism-invariant property. Q I < 1), so that G/[G,An] is a An- 15 Theorem 2.9: Let G be a finite solvable group. Then the following are equivalent: (1) G/F(G) has derived length a n. (ii) cofGS I S/corGS has derived length a n for all maximal subgroups S of G. (iii) C(n) is nilpotent. (n) (n) (iv) H «4‘4 G for all subgroups H of G. (v) H <4‘4 G for all proper abnormal subgroups H of G. 2322;: (i) w (ii) 4 (iii) is immediate from Lemma 2.8 and Corollary 2.2. (iii) a (iv) a (v) 4 (ii) follows as in the proof of Theorem 2.7, replacing yn( ) by ( )(n) and "nilpotent of class E n" by "solvable of derived length a n." H We have seen that if the cofactors of all the maximal subgroups of a finite group G are nilpotent, then G is solvable and of Fitting length at most 2; and in the preceding results, we have seen the ef- fect on G of requiring these cofactors to all be nilpotent of class at ‘most n. These conditions are not sufficient, however, to guarantee that G has a normal Sylow subgroup; in particular, the conclusions of Theorem 2.4 need not hold. In fact, it is not even sufficient to re- quire that the cofactors of all preper subgroups of G be abelian, as the following example shows. Example 2.10: Let G I S X A4, where 83 is the symmetric group on 3 3 letters and A4 is the alternating group of degree 4. Then: (i) G has no normal Sylow subgroups. GH is abelian for all prOper subgroups H of G. ‘ggggf: (i) is immediate since 83 has no normal 2-Sylow sub- (11) H/cor group and A4 has no normal 3-Sylow subgroup. 16 (ii) Let§ IS x<1>,X I<1>XA4,UIUX<1>Ithe 3 3 4 normal 3—Sylow subgroup of S5, and V'I <1l> x V I the normal 2-Sylow subgroup of A . We note first that G/F(G) I G/UV is isomorphic to 4 the direct product of a cyclic group of order 2 with a cyclic group of order 3, hence is abelian, so that from Corollary 2.2-(i), T/corGT is abelian for all maximal subgroups T of C. Now let H be a proper subgroup of G; we wish to show that H/corGH is abelian. Since this is trivially true if Hi4 G, we may assume that Hed G. Also, in view of the comment above and the fact that groups of order p or p2 (p a prime) are abelian, the only cases that need be checked are for |n| - 6, s, 12, or 13. Case 1: ‘H‘ I 6, H14 G.-—Let x I (a,b) E H have order 2, and z I (p,q) G H have order 3. Then at least one, but not both, of b and q is 1. For if b 9‘ 1 and q f 1, then ; for otherwise, H I H-S-3/ S3 I G/ S3 I A4, from which it follows that H has no normal subgroup of order 2; how- ever, H n A] is normal in H and of order 2. Now, suppose first that 2||H n 3|; let y I (a,1) E H n SS have order 2, and z I (p,q) E H have order 3. Then ap has order 2, and (1,q2) I (ap,q)2 G H. It follows then that q I 1, since otherwise we would have < (l,q2) , (l,b) > I X4 c H, which implies that H I A4 is normal in G. Thus, < (a,1) ,(p,l):> I 83 is contained in H, hence in GH is cyclic of order 2. Suppose now that ZI|H 0 S5], and thus 3“H 0 S5]. Then the nor- mal subgroup U.of G is contained in H, hence in corGH, so that H/cordH corGH, so that H/cor has order 4 and is therefore abelian. Case 4: ‘H‘ I 18, H‘d G;-—In this case, the normal subgroup U of G must be contained in H, hence in cor H, so that ‘H/corGH| § 6. G We may assume that ‘H/corGH‘ I 6 (since in the other cases H/corGH is clearly cyclic), and thus that U'I corGH. Then G/corGH I G/ll is isomorphic to C2 X A.4 where C2 is a cyclic group of order two. It is easily checked that this group has no proper subgroup isomorphic to S3; consequently, H/cor H cannot be isomorphic to S and is, therefore, a G cyclic group of order 6. fl 3 If we now require that in addition to the cofactors of all maximal subgroups of G being nilpotent, the proper normal subgroups of C also be nilpotent, we would certainly hape to be able to say more about the structure of G. Because of Example 2.13, we cannot hope to 18 recover all the results of the theorems of Schmidt, Iwasawa, and Rose (Theorems 2.3 and 2.4). Nevertheless, we do find that the structure of G is quite severely restricted, and that if G is not itself nilpotent, then, incomparison with Theorem 2.3, it is within a prime of being nilpotent; more precisely, G/F(G) is of prime order. To state the complete result, the following definitions are needed. Definition 2.7: A finite group G is said to be p-nilpotent if it has a normal p-complement; that is, there exists K‘4 G such that pX|K‘ and \G:K| is a power of p. It is quite easy to show as a consequence of this definition that G is nilpotent if and only if G is p-nilpotent for all primes p which divide |c|. Definition 2.8: A subgroup H of a given finite group G will be said to be nearly normal Lg §,if cofGH I H/corGH is trivial or of prime order. H will be said to be nearly subnormal ig_§ if scofGH I H/scorGH is trivial or of prime order. Theorem 241: Let \G‘ I Ill p:1 where the pi are distinct primes iIl dividing |G|. Suppose that cof S I S/corGS is nilpotent for G all maximal subgroups S of G and that all proper normal sub- groups of G are nilpotent, but that G itself is not nilpotent. Then the following hold. (a) G is solvable. (b) F(G) is the unique maximal normal subgroup of G. (c) There exists a prime, say pr, dividing |G| for which the following hold. (1) G is pr-nilpotent. (2) For all i I r, G is non-pi-nilpotent. 19 (3) |c:r(c)| = pr- (4) For any proper subgroup H of G, scofGH I H/scorGH has order 1 or pr; in particular, all proper sub- groups of G are nearly subnormal in G, and all maximal subgroups are nearly normal in G. (5) For each i f r, G has a normal pi-Sylow subgroup Pi' (6) For each i I r, Pi c G'; thus, G/G' is a pr-group. a (7) p divides d I H (p.J -l) for all i f r. r pi 1‘1 1 (8) For each pr-Sylow subgroup Q of G, QG I G, that is, Q has G as its normal closure. (9) For each i f r such that Pi is abelian, (i) G has exactly pia1 distinct pi-complements, (ii) CG(P1) I F(G), and thus G induces in P1 a cyclic group of automorphisms of order pr, (iii) the number of pr-Sylow subgroups of G is a a1 multiple of pi. (10) If Pi is abelian for all i I r, then: (i) G has exactly ‘Gl/pr at distinct pr-Sylow subgroups, each of which is abnormal in G. (ii) The set of pr-Sylow subgroups of G I the set of system normalizers of G I the set of Carter subgroups of G. (iii) If, in addition, ar I 1, then G I X U X' where X is the set of pr-elements of G and X' is the set of pé-elements; thus, G is a Frobenius group with kernel I F(G) I G' I the normal pr-complement of C; also, Z(G) I. 20 £293: (a) follows from Theorem 2.5; and (b) is inmediate since every proper normal subgroup of G is, by hypothesis, nilpotent and hence is contained in F(G). (c) (3) Since G is solvable and F(G) is a maximal normal sub- group of G, G/F(G) is of prime order. Relabelling if necessary, we may assume that ‘G:F(G)‘ I pr. (1) We have \G:F(G)‘ I pr’ and F(G) is nilpotent, hence pr- nilpotent. Let T be the normal pr-complement of F(G). Then T is characteristic in the normal subgroup F(G) of C so that T14 G; also, a ‘G:T| I ‘G:F(G)HF(G):T| I pr r . Thus T is a normal pr-complement of G. (4) Let H be any proper subgroup of G. If H : F(G), then by the nilpotence of F(G), H 4 4 F(G) 4 G, so that H 4 4 G. Thus scorGH I H and |H/scorGH| I 1. Now suppose H ¢ F(G) I F. Then by the maximality of F, we have HF I G so that [H n Fl I 1%J- : %+.-il’§i I EL. r pr From the nilpotence of F, we have H n F 4 4 F 4 G, that is, H n F 4 4 C; it follows that H n F c scorGH. Hence, since ‘H:H n F‘ I pr, H/scorGH has order 1 or pr' In either case, therefore,|scofGH| I l or pr as we wished to show. If S is a maximal subgroup of G, then by Lemma 1.3, scorGS I corGS. By what we have just shown, S/corGS I S/scorGS has order 1 or pr so that S is nearly normal in G. (5) If i f r and Pi is a pi-Sylow subgroup of F(G), then by 1 char é F(G) 4 G, hence P1 4 G. And since ‘G:F(G)‘ I pr is prime to p1, P1 is a normal pi-Sylow sub- the nilpotence of F(G), we have P group of G. (2) Suppose that for some i f r, G is pi-nilpotent. Then a there exists a normal subgroup T1 of G with ‘G:T1\ I pi 11. By hypothesis, T1 is nilpotent, and thus has a characteristic pr-Sylow 21 a. subgroup Q. Then Q 4 G; and since lG:T‘ = pi 1 is prime to pr, Q is a normal pr-Sylow subgroup of G. However, this together with (5) implies that all the Sylow subgroups of G are normal in G, contradict- ing the nonnilpotence of G. Therefore, G is non-pi-nilpotent for each i I r. (6) Let G29.) denote the smallest normal subgroup of G for which the factor grbup is an abelian pi-group. We show first that ' I G. For suppose that G' I G for some i I r. (pi) (pi) Then G' is a proper normal subgroup of G, hence is nilpotent by (pi) hypothesis, and in particular, is pi-nilpotent. This means that there for i I r, G (pi) ‘G(pi):Ti‘ = a power of pi. Then Ti 4 G and ‘GzTil = \G:G(pi)\\c(pi):Ti\ is a power of pi so that G is pi-nilpotent, in contradiction to (2). exists a characteristic subgroup T1 of G with Ti 3 pi-group and Thus, I G for each i I r. From one of the basic transfer 6' (pi) theorems (see, for example, Scott [19], 13.5.2), it follows that Ill <‘1> = G/G(pi) that is, Pi C G'. Pi/Pi n G'. For each i I r, therefore, P1 0 G' I Pi’ (7) Case 1: For some i I r, the normal pi-Sylow subgroup P1 of G is not minimal normal in G.I—Iet Miq G with I M‘i Pi; we show that G/M is not pi-nilpotent. For suppose that it is; then there 8..'lII. m. exists T14 G with |G:T| = ‘G/MzT/M] = pi " where |M\ = pi. 1. Since M is properly contained in P pi divides ‘G/M‘ so that T is i, a prOper normal subgroup of G, hence is nilpotent by hypothesis; in particular, T is pi-nilpotent. There exists, therefore, a character- m istic subgroup U of T with lrzul = pi 1. Then U 4 c and ‘G:U‘ = a, ‘G:T‘]T:U] I pi 1’, which means that U is a normal pi-complement of G. This, however, contradicts (2). 22 Thus, G/M is not nilpotent. And since all proper normal sub- groups of G are nilpotent, the proper normal subgroups of G/M are all nilpotent. Also, the cofactors of the maximal subgroups of G/M are nilpotent. For if S/M is a maximal subgroup of G/M, then S is a maximal Subgroup of G. By hypothesis, cofGS is nilpotent so that, by Lemma 1.4, cofG/M(S/M) I cofGS also is nilpotent. The hypotheses, therefore, hold for G/M; consequently, since each pi divides ‘G/M‘, we have by induction that for each i I r, pr 9: b1 j b1 br divides d = H (p. -l) where ‘G/M] I p ... p . Hence, p also P- -_ 1 1 r r 1 j-l ai . divides d I H (p J -l) for each i I r. Case 2. ar > l.-—Let Q be a pr-Sylow subgroup of G. From (4) we have ‘Q/corGQ| I pr, and thus, since ar > 1, corGQ I < l>. Now suppose that G/corGQ is pi-nilpotent for some i I r. Then there a a . . . _ i _ exists Ti 4 G Wlth ‘G/corGQ . Ti/corGQ| - pi so that lG:T| - pi i a which implies that G is pi—nilpotent. This, however, contradicts (2). Therefore, G/cordQ is not nilpotent; and as in Case 1, all proper normal subgroups of G/corGQ and all cofactors of maximal sub- groups of G/corGQ are nilpotent. Hence, by induction, since all the pi divide ‘G/corGQ‘, we have the result as in Case 1. Case 3: ar I l and for all i I r the pi-Sylow subgroup Pi of G is minimal normal in G.-—In this case, each Pi for i I r is elemen- e. ' ° = 1 = - tary abgiian, thus lAut(Pi)‘ pi .dpi where ei a1(ai l)/2 and dp = r1 (pi J - 1). Now from (9), which will be proved independently i j=1 of (7), we have CG(Pi) I F(G) for all 1 I r, so that ‘G:CG(P1)‘ I pr for i I r. Since G/CG(Pi) is isomorphic to a subgroup of Aut(Pi), this means that pr divides ‘Aut(Pi)‘, and hence divides dp for all i 23 i I r. (8) Suppose that QG i G for some pr-Sylow subgroup Q of G. Then QG is a prOper normal subgroup of G, hence is nilpotent so that Q is characteristic in QC. But this implies, since QGT4 G, that Q is normal in G, which together with (5) means that all the Sylow sub- groups of G are normal in G, contradicting the nonnilpotence of G. Therefore, QG I G for each pr-Sylow subgroup Q of G. (9) (i) This is an immediate consequence of Theorem 2.25: Given a solvable non-p-nilpotent group G having all proper normal subgroups p-nilpotent, and with ‘G‘ I pam where (p,m) I 1. If G has an abelian normal p-Sylow subgroup, then G has exactly pa distinct p- complements. (ii) Let Pi be abelian for some i I r. Since Pi C F(G) and F(G) is nilpotent, it follows that F(G) centralizes Pi' Thus, F(G) C CG(Pi); from the maximality of F(G), we have CG(Pi) I F(G) or C. Now, Pi ¢ Z(G); otherwise, Pi would centralize and hence normalize a p.-complement T. of G, which would imply that T is normal in G, con- 1 1 i tradicting (2). Therefore, CG(Pi) I G so that CG(Pi) = F(G). (iii) Let Pi be abelian for some i I r; then, from (i), G has 3. exactly pil' distinct pi~complements. We show first that each pr-Sylow subgroup Q of G is contained in some pi-complement. For this, we have F(G) nilpotent, hence p-nilpotent, and thus has a unique pi~complement a. W which is normal in G. Since ]G:F(G)] I pr and \F(G):W‘ I pi 1 , it a -l a. follows that [W] I p I. H P. J. - r . . j JI1.r Now, since W 4 C, Q n W is a pr-Sylow subgroup of W so that a -1 Q n W has order pr 1r . Therefore P W a W r ' l 1 Q n w p gr-1 r‘ 1 inPJ 24 which shows that QW is a pi-complement of G. We note next that p:hr divides the order of each pi-complement of G for i I r. [Consequently, each pi-complement of G, for i I r, con- tains a pr-Sylow subgroup of G. Also, no two distinct pi-complements X I Y of G can contain the same pr-Sylow subgroup of G. For since F(G) is normal in G, X 0 F(G) and Y n F(G) are pi-complements of F(G); thus, X n F(G) = Y n F(G) = W, the unique pi-complement of F(G). Now, a. G /p.1 |x:w| = l ‘ 1 Hi = \G:F(G)\ = p lF|/pi 1' and similarly, lY:W\ I pr. Therefore, since W c X n Y and X n Y I X, Y, we must have X n Y I W. Since prar does not divide ‘WI I ‘Xn Y], it follows that X n Y contains no pr-Sylow subgroup of C so that X and Y can have no pr-Sylow Subgroup of G in common. Finally, since G is solvable, any two distinct pi-complements X and Y of G are conjugate. Consequently, X and Y must contain the Same number, say 1, of pr-Sylow subgroups of G. We therefore have the following: each pr—Sylow subgroup of G is contained in some pi-complement of C; no two distinct pi-complements have any pr-Sylow subgroup of G in common; each pi-complement of G con- tains X pr-Sylow subgroups of G; G has exactly p:H' distinct pi- complements. From these it follows that G has exactly kpiai distinct pr-Sylow subgroups. (10) (i) Let n I the number of distinct pr-Sylow subgroups of G. Since Pi is abelian for all i I r, from (9)-(iii) we have that a. b a pil"n for all i I r, so that n I p I- H p r ihi ever, from the Sylow theorems, n s 1 (mod pr). This implies that for some br 2 0. How- 25 a. a l r br = 0, and hence n = II pi = ‘G‘ I1)r . ifr Now, if Q is any pr-Sylow subgroup of G, by what we have just shown, IG:Q| = the number of distinct pr-Sylow subgroups of G, which in turn is equal to ‘G:NG(Q)‘ . Hence, Q = NG(Q) so that Q is abnormal in G, since the normalizers of Sylow subgroups are abnormal subgroups. (ii) Since F(G) is nilpotent as in clue), it follows by a result of Carter [3] that the Carter subgroups of G are identical with the system normalizers in C. Now, G has a unique pr-complement since it is pr-nilpotent; and for each i 1‘ r, G has pia1 distinct pi-complements. Thus G has |G‘/p:r distinct Sylow (complement) systems, and hence has at most. |G‘/p:r distinct system normalizers. But each pr-Sylow subgroup is a nilpotent self-normalizing subgroup of G, that is, a Carter subgroup of G, and by the preceding comments, is, therefore, a system normalizer of G. It now follows that since G has exactly |G‘/p:lr distinct pr- Sylow subgroups, these must be all the system normalizers of G. (iii) If ar = 1, then by (i), G has exactly ‘G‘Ipr distinct pr-Syluw subgroups. Since each of these has order pr and vvvry pr- element of G belongs to some pr-Sylow subgroup, the number nr of non- identity pr-elements of G is given by nr = £§L(pr - l) = ‘G‘ - £§L . r Since G is p -nilpotent, the number n; of pé-elements different from r l is n - %§L - 1. Since nr +’n; ' ‘G‘ - l, the first conclusion of r | r (iii) now follows. From (1) we have Q NG(Q) for each pr-Sylow subgroup Q of G. Since \Q| . pr, this implies that Q n Q" - < l> for all x e c - Q, and hence that G is a Frobenius group with kernel = the set of pL-elements of G, which by (3) is equal to F(G). Also, from (6), G/G' is a pr- group so that ‘G/G" - l or pr' Since G is solvable, 6' f G, hence 26 ‘G/G" ' pr. Therefore, since 6' c F(G) and ‘G:F(G)\ = pr, we have G' - F(G). Finally, it follows from a well-known prOperty of solvable groups having all Sylow subgroups abelian (Taunt [20], also proved by Huppert in [11], 14.3) that G' n Z(G) "; thus, ‘Z(G)‘ ' l or pr since |G/G" - pr. But since the pr-Sylow subgroups of G are non- normal, we have |Z(G)| 1‘ pr so that Z(G) ' < 1). [1 Statement (c)-(4) of the preceding theorem shows that if we extend the requirement of nilpotence of normal subgroups to the nearly normal maximal subgroups of G, we then have all the maximal subgroups of G being nilpotent so that Theorem 2.3 holds. This gives the following corollary to Theorem 2.11. Corollary 3;lg: Suppose that the cofactors of all maximal subgroups of G are nilpotent as are the nearly normal maximal subgroups of G, but that G itself is not nilpotent, say G not p- nilpotent. Then the following hold. (1) All proper subgroups of G are nilpotent. (ii) ‘6‘ = paqb for some prime q f p, and the conclusions of Theorem 2.3 hold. (iii) The conclusions of Theorem 2.11 hold with r = 2, p1 = p, p2 ' q, a1 = a, and a2 = b. The condition imposed in Theorem 2.11 that a nonnilpotent group G have the cofactors of all its maximal subgroups and all its normal subgroups nilpotent does not impose any bounds on ‘n(G)‘ . the number of distinct prime factors of ‘6‘. Neither does it guarantee that G has a normal Sylow subgroup for which the factor group is nilpotent. In fact, it is not even sufficient to require that the cofactors of all 27 proper subgroups and all normal subgroups of G be cyclic. More pre- cisely, we have the following example. Example gglé; For all n e 3, there exists a finite nonnilpotent group G such that: (i) the cofactors of all proper subgroups of G and all prOper normal subgroups of G are cyclic; (ii) ‘6‘ is divisible by n distinct primes; (iii) G has no normal Sylow subgroup'Q for which GflQ is nilpotent. 2332:: Let 2 = {p1,... , pn_1} be any collection of n-l dis- tinct odd primes and P "<}{ > be a cyclic group of order pi for each i. i 1 Then each Pi has an automorphismai of order 2. Let K = P1 X... x Pn- a ==a1 X ...X an_1 is an automorphism.of K of order 2. Now let G be 13 the extension of K by a. G is not nilpotent; for if it were, then. would be a normal subgroup of G, hence would centralize K, which con- tradicts the fact that |al = 2. We show now that G satisfies the three conditions. Since (ii) is clear, only (i) and (iii) require proof. (1) Let H be any proper subgroup of G. Suppose first that ZIIH‘, say [3| x pi ...p where each pi 6 2. In this case, we must 1 1k have R 5 P. X... X P which is cyclic. 11 1k Suppose now that 2||H|. Then since H f G, some ptY‘H‘ so that Pt n H "<:1>u Now, H is not normal in G; for suppose that it is. Then since 2||H‘, the 2-Sylow subgroups of G are contained in H; in t axta E H. But since Pt - is a characteristic particular, a 6 H. Since H is normal in G, we have x laxt E H, and hence, [xt,a] ' xt subgroup of G, we have [xtm] 6 Pt' Thus, [live] 6 Pt n H . < 1), which implies that Cat)“ - xt' This, however, contradicts the fact that a t .- “‘P has order 2. Therefore, H A G; and it is clear that t 28 corGH is the product of the Pi for pi dividing \H‘. Consequently, H/corGH is cyclic of order 2. (iii) The P are the normal Sylow subgroups of G. If for some i i, G/Pi is nilpotent, then TP is a proper normal subgroup of G for T i a 2-Sylow subgroup of G. However, this contradicts the proof in (i) that if H is a prOper subgroup of G and 2||H|, then H is not normal in G. Therefore, G has no normal Sylow subgroup for which the factor group is nilpotent. H 2.3 p-nilpotent Cofactors or Subcofactors We now turn to a consideration of those finite groups G for which the cofactors or subcofactors of certain proper subgroups of G are p-nilpotent, that is, they possess a normal p-complement. As in the preceding section, we will later require that the proper normal subgroups of G also be p-nilpotent in order to further delimit the structure of G, and finally, that the somewhat normal subgroups of G also be p-nilpotent. One of the major results that we seek to extend is the classic theorem due to Ito [13] (also proved by Huppert in [11], 5.4). Theorem 2.14: If all the proper subgroups of a finite group G are p—nilpotent, but G itself is not, then (i) all proper subgroups of G are nilpotent. Thus, the conclusions of Theorem 2.3 hold; that is, (11) IGI - paqb for some prime q 1‘ p; (iii) G has a normal p-Sylow subgroup P; P has class s 2, and in fact, §(P) c Z(G); if p is odd, exp(P) = p, and if P - 2: exP(P) i 4; 29 (iv) the q-Sylow subgroups of G are cyclic; and if Q is any such, 6(Q) : Z(G). Rose [17] has also established some results in this direction. The following are two such theorems. Theorem 2.15: If every proper self-normalizing subgroup of G is p-nilpotent, then G has a normal p-subgroup PO (which may be trivial) such that G/Pb is p-nilpotent. Theorem 2.16: If every prOper abnormal subgroup of G is p-nilpotent and either p is odd or the p-Sylow subgroups of G are abelian, then the conclusion of Theorem 2.15 holds. That the added conditions on p in the preceding theorem cannot be omitted is shown by the following example of Rose, the details of which appear in [17]. Example 2:11: Let H be the simple group of order 168 (H 3 PGL(3,2) = GL(3,2) = PSL(3,2)), and G the split extension of H by the automorphism¢y of R defined by a: x.» (x-1)t, where yt denotes the transpose of the matrix y. Then every proper abnormal subgroup of G is supersolvable, hence 2-nilpotcnt, but G is not solvable, hence not 2-solvab1c. As we now show, the two results of Rose (Theorems 2.15 and 2.16) can be extended by requiring not that the self-normalizing or abnormal subgroups themselves be p-nilpotent, but only their subcofactors. To establish this, we will use the following well-known results due to Burnside [2] and the following lemma. Theorem 2.18: (1) If the finite group G is not p-nilpotent, then G has a nontrivial p-subgroup Pb and a p'-element x such that x E NG(P0) but x f CG(PO). 30 (2) If G is a finite group with P a p-Sylow subgroup, and if P c Z(NG(P)), then G is p-nilpotent. Lemma 2.19: Let 9 be a group-theoretic preperty such that products of normal 9-subgroups of a group are again e-groups. If the finite group G has a nontrivial subnormal 6-subgroup, then G has a nontrivial normal 9-subgroup. Proof: Let K f < 1> be a e-subgroup of G, and K=KO$K1$K2$ ...?Kr_1?1(r =G where r is the minimal length of subnormal chains from K to G. We use induction on r. The result is trivially true if r e 1; thus, suppose r > 1. By induction, Kr-l has a nontrivial normal 9-subgroup. Let K* be the product of all such. Then K* is a nontrivial e-group, is clearly characteristic in Kr-l’ and hence is normal in G. n Definition 2.9: A finite group G is said to be pfsolvable if it has a normal series l= KO C K c...c K.n = G in which each 1 factor Ki/Ki-l is either a p-group or a p'-group. For a p-solvable group G, the ascending p-series , = C i - = <1» P0 NorPlc—NICPZC ('PCCJNt G is defined by taking Ni/Pi to be the largest normal p'- subgroup of G/Pi’ and P /N1 the largest normal p-subgroup i+1 of G/Nio The £fl£28£h_2£_§3 LP(G), is the least integer t such that Nt ' G. The following are well-known consequences of this definition: (1) G p-nilpotent a G is p-solvable. (2) G is solvable if-f G is p-solvable for all primes p which divide |c| . 31 (3) LP(G) is the smallest number of p-factors that can occur in a normal series of G for which the factor groups are either p-groups or p'-groups. Definition 2,19: For a subgroup H of a finite group G, the 21225' nomlizer g n in Q, denoted N: (H), is defined to be the sub- group in which the ascending chain H = H d H 4 H 4 ... , O l 2 defined by H i = NG(Hi-1)’ terminates. Theorem 2.20: If the subcofactor H/scorGH of each proper self- normalizing subgroup H of G is p-nilpotent, then there exists a normal p-subgroup P0 of G (Pb may be trivial) such that G/Pb is p-nilpotent. In particular, G is p-solvable of p-length § 2. ‘ggggf: The proof is by induction on ‘G‘. We may assume that G is not p-nilpotent, since the result is trivially true otherwise. It suffices to show that G has a nontrivial normal p-subgroup P0. For if H.“ H/Po is any proper self-normalizing subgroup of 5" G/Po, then clearly H is a proper self-normalizing subgroup of G; thus, by hypothesis, H/scor H is p-nilpotent; hence, by Lemma 1.4, so also is G H/scor-C-(H) a H/s corG that, by induction, there exists a normal p-subgroup P H. The hypotheses therefore hold for G'= G/Pb so 1 - P1/Pb of G such that G/ P1 is p-nilpotent. Then P is a normal p-subgroup of G l and G/P1 a GI P1 is p-nilpotent. Since G is not p-nilpotent, it follows by Theorem 2.18 that G has a nontrivial p-subgroup P and a p'-e1ement x such that x E NG(P)-CG(P). Let N " N; (P) be the hypernormalizer of P in G. If N ' G, we then have f Pro a G so that from Lemma 2.19, G has a nontrivial normal p-subgroup. The result then follows from our comments above. 32 Thus, suppose that N 9‘ G. Since N is clearly a self-normaliz- ing subgroup of G, we then have, by hypothesis, that N = N/scorGN is p-nilpotent. Now let P = P scorGN/ scorGN and §' = x scorGN . Since x normalizes P, we have P 4

; and since

is p-nilpotent as a subgroup of N, we also have <§> 4 . Since x is a p'- element of G, it follows that

= P X <§>, hence that ; centralizes‘P. Since x 4 CG(P), there exists u E P such that [u,x] = u'lx”lux # 1. And since ;'centralizes P; we have [u,x] E scorcN; also, [u,x] E P since x E NG(P). Thus 1 # [u,x] E P n scoréN so that P11 scoréN is a nontrivial p-subgroup of G; and since P‘4 4‘N, we have P{1 scorGN‘4‘4 scorGN'4'4 G, hence, P n scorGN 4‘4 G. From Lemma 2.19, we conclude that G has a nontrivial normal p-subgroup; and the result now follows as above. That G has p-length i 2 is now an immediate consequence of the remark (3) following Definition 2.9. For we have shown that G has a normal p-subgroup PO (perhaps trivial) such that G/P0 is p-nilpotent. Letting T/Po be the normal p-complement of G/PO, we have the normal series G :2 T :2 Po :2 < 1) with factors that are either p-groups or p'-groups, at most two of which are nontrivial p-groups. fl Corollary 2.21: If the subcofactor H/scor H of each proper self- G normalizing subgroup H of G is 2-nilpotent, then G is solvable, and there exist normal subgroups H, K of G such that H/K is isomorphic to a 2-complement of G. Proof: From the theorem, there exists a normal 2-subgroup K of G such that G/K is 2-nilpotent. Thus there exists H‘4 G such that H/K is a 2-complement of G/K (and hence is isomorphic to a 2-complement 33 of G since K is a 2-group). Now, H/K has odd order. By the Fcit- Thompson Theorem, therefore, H/K is solvable. Since the 2-groups K and G/H are solvable, it follows that G is solvable. H To extend Theorem 2.17, we make use of the Glaubermann- Thompson Theorem concerning the Thompson subgroup J(P) of a p-Sylow subgroup P. Various definitions of J(P) have been given; we shall use the following (given by Gorenstein in [8], in which a proof of the Glaubermann-Thompson Theorem also appears). Definition 2.11: For a given p-group P, the Thompson subgroup J(P) of P is defined by J(P) = , where d(P) is the collection of all abelian subgroups of P of maximal order. N953: For P and J(P) as in the definition, we have Z(P) C Z(J(P)). For if A E d(P) and x 6 Z(P), then is abelian; thus, by the maximality of ‘A‘, we have x E A, from which this in- clusion follows. Theorem 2.22: (Glaubermann-Thompson Theorem) Let P be a p-Sylow sub- group of the finite group G with p odd. Then, if NG(Z(J(P))) is p-nilpotcnt, so also is G. Theorem 2.23: If the subcofactor H/scorGH of each proper abnormal sub- group H of G is p-nilpotent and either p is odd or the p-Sylow subgroups of G are abelian, then there exists a normal p-sub- group PD of G (Pb may be trivial) such that G/P0 is p-nilpotent. In particular, G is p-solvable of p-length é 2. Proof: The last statement follows as in the proof of Theorem 2.20; thus only the existence of PO requires proof. For this, we pro- ceed by induction on ‘G‘. Suppose that G has a nontrivial normal p- subgroup P3. If fi'= H/P; is any proper abnormal subgroup of G'= G/P*, clearly H is a proper abnormal subgroup of C so that, by hypothesis, 34 H/scorGH is p-nilpotent, hence so also is HVScorC(H) E H/scorGH (by Lemma 1.4). Since the other hypotheses obviously hold for G, we have by induction that 5 possesses a normal p-subgroup P6 = Po/Pg such that G/ P0 is p-nilpotent. Then P is a normal p-subgroup of G and 0 is p-nilpotent. Thus, we may assume that G has no non- -- ~ 0 = G/lq) trivial normal p-subgroup, and hence, by Lemma 2.19, no nontrivial G/P subnormal p-subgroup; and we must show that G is p-nilpotent. Let P be a p-Sylow subgroup of G. We consider first the case where P is abelian. Let N = NG(P). Then N is an abnormal subgroup of N is - G P GN be the normal p-complement of N/scordN. Now, since P14 N, we have P n scorGN 4 scorGN 4 4 G so that P11 scorGN 4 4 G. Since G has no nontrivial subnormal p-subgroups, we have P n scoréN = and hence that scordN is a p'-group. This implies then that T is a G, and since P16 G, N f G; by hypothesis, therefore, N/scor nilpotent. Let T/scor normal p-complement of N, from which it follows that N = P x T. There- fore, T : CG(P); and since P is abelian, P : CG(P). From this it follows that P C Z(NG(P)) so that, by Theorem 2.18, G is p—nilpotent. Now consider the case where p is odd, and let S = NG(Z(J (P))), where J(P) is the Thompson subgroup of P. Since Z(J(P)) is char- acteristic in P, we have Z(J(P)) 4 NG(P), and thus NG(P) C N. Since NG(P) is abnormal in G, it follows that N also is abnormal in G. And since G has no nontrivial normal p-subgroups, N f G so that N is a proper abnormal subgroup of C. By hypothesis, therefore, N/scOr N is G p-nilpotent. Suppose now that P1 = P n scordN #‘<:1>u Then, since scoréfi 4 N, we have P1 = P!) scor¢N14 P, and hence P1 must intersect Z(P) nontrivially. Therefore, since Z(P) C Z(J(P)) by the above note, 35 this means that P = P n Z(J(P)) 1‘ < l> also. 2 1 Now, 92 = P1 n Z(J(P)) = P n sootGfi n 2002)) = Z(J(P))) n scorGN. ‘2'. ~ 5:: And since Z(J(P))‘d N, we have P2 = Z(J(P)) n scorCN 4 scorGN‘4 4 G, so that P 1414 G. Thus P is a nontrivial subnormal p-subgroup of G, 2 2 which is a contradiction to the fact that G has no such subgroup. Consequently, P n scorGN = tso that scordfi is a p'-group. Since N/scorGN is p-nilpotent, it now follows, as in the previous case, that N = NG(Z(J(P))) is p-nilpotent. By the Glaubermann-Thompson Theorem, therefore, G is p-nilpotent. fl Corollary 2.24: If the subcofactor H/scor H of each proper abnormal G subgroup H of G is 2-nilpotent and the 2-Sylow subgroups of G are abelian, then G is solvable, and there exist normal sub- groups H, K of G such that H/K is isomorphic to a 2-complement of G. Ppggf: This follows from the preceding theorem and the Feit- Thompson Theorem in the same manner as Corollary 2.21 was proved. fl Example 2.10 of the preceding section shows that a non-p-nil- potent group having the cofactors of all its proper subgroups p-nil- potent need not have a normal p-Sylow subgroup. This is no longer the case if we require that, in addition, all the prOper normal subgroups of G be p-nilpotent. Before establishing this and other properties of G, however, we first prove the following result (which we have already used in Theorem 2.11). Theorem 2.25: Let G be a solvable non-p-nilpotent group having all proper normal subgroups p-nilpotent, and let ‘G‘ = pam where (p,m) = 1. If G has an abelian normal p-Sylow subgroup P, then G has exactly pa distinct p-complements. 36 Proof: Extend G :3 P :3 < l> to a chief series = ... = :3 ... = 11> G G013 :3 Gm PJD Gm+l ID Gm+n < and set Pi = Gm+i for i = 0, 1, ..., n. We assert first that for each i e l, G/Pi is not p-nilpotent. For suppose that for some i E l, G/Pi is p-nilpotent, say with normal p-complement K/Pi' Then K.4 G; and since Pi<; P for i a l, p“G/Pi‘ so that K is a proper normal subgroup of G. By hypothesis, therefore, K is p-nilpotent. If T is the normal p-complement of K, then T char é K‘4 G, which implies that T14 C; also, ‘GzT‘ = ‘c:1<“1<:'r‘ = ‘G/Pi:K/Pi“K:T‘ = pa. Thus T is a normal p- complement of G; but this contradicts the non-p-nilpotence of G. Let now (G/Pi)' denote the least normal subgroup of G/Pi (p) for which the factor group is an abelian p-group. Then (G/Pi)(p) = G/Pi for all i g 1. For suppose not and that (G/Pi)(p) = L/Pi where L‘g G for some i g 1. Then, by hypothesis, L is p-nilpotent, and hence so also is L/Pi. Let U/Pi be the normal p-complement of L/Pi. Since U/Pi is a characteristic subgroup of L/Pi, we have U/Pi 4 G/Pi; also, ‘G/Pim/Pi‘ = ‘G/PizL/Pi“L/Pi:U/Pi‘ is a power of p. This means that G/Pi is p-nilpotent, a contradiction to what we have shown above. Now, for each i E 1, let Ti be the transfer of G/Pi into its abelian normal p-Sylow subgroup P/Pi' From the basic properties of the transfer (see, for example, Scott [19], 13.5.2, 13.5.5), we have that ker Ti = (G/Pi)(p)’ and since P/Pi is an abelian normal p-Sylow Subgroup of G/Pi’ Ti(G/Pi) = (P/Pi) n Z(G/Pi). Since we have just shown that (G/Pi (p) = G/Pi for each i a l, we have Ti(G/Pi) = < 1), and hence that (P/Pi) n Z(G/Pi) - < 1> for all i .1: 1. For each i = l, 2,... , n, therefore, since P1_1 : P and (P/Pi) n Z(G/Pi) - < 1), we have (Pi_1/P1) n Z(G/Pi) - < l>. In 37 alternate terminology, this means that each of P/Pl’ Pl/PZ’ ... , Pn-l/Pn is an eccentric chief factor of G. A result of P. Hall [10] states that for a solvable group G, the number of p-complements of G is equal to the product of the orders of the eccentric p-chief factors of G. Thus we have that the number of distinct p-complements of G is equal to iEI‘Pi'llpi‘ - ‘P‘ a pa. H We now examine the structure of a non-p-nilpotent finite group G having all its proper normal subgroups p-nilpotent as well as the sub- cofactors of its self-normalizing or abnormal subgroups. Although Example 2.13 shows that we cannot hope to recover all of Theorem 2.14, we do discover a considerable amount of structure in G. Theorem 2.26: Let G be a finite non-p-nilpotent group having all of its proper normal subgroups p-nilpotent, and let ‘G‘ - pam with (p,m) I 1. Suppose also that one of the following two conditions holds: (a) The subcofactor of each preper self-normalizing subgroup of G is p-nilpotent. (b) The subcofactor of each proper abnormal subgroup of G is p-nilpotent and either p is odd or the p-Sylow subgroups of G are abelian. Then, the following hold: (1) G has a normal p-Sylow subgroup P. (ii) P c G', and thus G/G' is an abelian p'-group. (iii) FP(G), the largest normal p-nilpotent subgroup of G, is the unique maximal normal subgroup of G, and G/FP(G) is a p'-groupo (iv) For all KTg G, (‘G/K‘,dp) f l where dp - a (p1 - 1). i=1 38 (v) If P is abelian, then CG(P) = Fp(G); thus G induces a p'-group of automorphisms in P. (vi) If G is solvable, then ‘G:FP(G)‘ = q for some prime q dividing ‘c‘ and dp. If also 2 is abelian, then G has exactly pa distinct p-complements. Ppggf: (1) Suppose that G has no normal peSylow subgroup. From Theorem 2.20 or 2.23, there exists a normal p-subgroup Pb of G such that G/Po is p-nilpotent, say with normal p-complement K/Po. Since P0 is not a p-Sylow subgroup of G, we have K‘g G. By hypothesis, therefore, K is p-nilpotent, say with normal p-complement T. Since T is characteristic in the normal subgroup K of G, we have T14 G; also, ‘G:T‘ = ‘G:K“K:T‘ = ‘G/PozK/PO‘IK:T‘ is a power of p. However, this means that T is a normal p-complement of G, contradicting the non-p-nilpotence of G. Therefore, G does have a normal p-Sylow subgroup. For (ii), the proof of part (c)-(6) of Theorem 2.11 carries over with pi = p. (iii) is immediate. For if K is any proper normal subgroup of G, then, by hypothesis, K is p-nilpotent and thus is contained in FP(G), from which it follows that Fp(G) is the unique maximal normal Subgroup of G. In particular, the normal p-Sylow subgroup P of G must be con- tained in Fp(G) so that G/Fp(G) is a p'-group. (iv) We suppose this result to be false, and let G be a minimal counterexample. Then there exists L‘$ G with (‘G/L‘,dp) = 1. Let K be a maximal normal subgroup of G containing L and let ‘G:K‘ = n; than n also is relatively prime to dp. Now, from (iii), FP(G) is the unique maximal normal subgroup of G. Consequently, we have K ' FP(G). And from (iii) again, FP(G) has index prime to p. Therefore, G/K is a 39 p'-group so that ‘G:K‘ I n is prime to p also, and the normal p-Sylow subgroup P of G is contained in K. Suppose now that P is not minimal normal in G. Then there exists M 4 G with < 1> I M s P. If G/M were p-nilpotent, say with normal p- complement U/M, then U14 G, and U I G since p“G/M‘; by hypothesis, therefore, U is p-nilpotent. Then the normal p-complement V of U is normal in G, and ‘G:V‘ I ‘G:U“U:V‘ I ‘G/M:U/M“U:V‘ is a power of p so that V is a normal p-complement of G. But this contradicts the non- p-nilpotence of G. Thus, G/M is not p-nilpotent. Also, the proper normal subgroups of G/M are clearly p-nilpotent. Since, from Lemma 1.4, scofcfuifilu) I scofGH for each prOper subgroup H/M.of GIM, and since it is immediate from the definitions that H is self-normalizing (abnormal) in G if'HfM is self-normalizing (abnormal) in G/M; it follows that (a) or (b) holds in G/M.according as (a) or (b) holds in G. In addition, x/u 3 G/M since xfl c, and ‘G/M:K/M‘ - ‘9le - n is prime to dp, hence is prime to d: I i§1(pi - l) where p8 is the highest power of p which divides ‘G/M‘ . But this means that G/M is a counterexample to this result with ‘G/M“< ‘G‘, which contradicts the minimality of G. Therefore, P is a minimal normal subgroup of G, hence is elementary abelian. The order of the automorphism group of P, Aut(P), is thus equal to pe.dp where e I a(a - 1)/2. Now, since T is a normal p-commlement of K and P c K, we have K I P X T so that T centralizes P; and since P is abelian, P’c GG(P) also. Thus, K.I P X T : CG(P) from which it follows that the order of G/GG(P) divides ‘G:K‘ I n and hence is prime to both p and dp. How- ever, G/GG(P) is isomorphic to a subgroup of Aut(P) so that ‘G/GG(P)‘ must divide ‘Aut(P)‘ I pe.dp. Thus we have a contradiction so that no such minimal counterexample to (iv) can exist. 40 (v) FP(G) is, by definition, the largest normal p-nilpotent subgroup of G; let W be its normal p-complement. Since P is normal in G and is trivially p-nilpotent, we have P C Fp(G). It follows that FP(G) I P X W so that W centralizes P. Since P is abelian, P ; CG(P) also, and hence Fp(G) C CG(P). From Theorem 2.18, since G is not p-nilpotent, P ¢ Z(G) I Z(NG(P)). Thus, by the maximality of Fp(G) established in (iii), we have Fp(G) I CG(P). (vi) For G solvable, we have ‘G/Fp(G)‘ I q for some prime q, since Fp(G) is a maximal normal subgroup of G; and from (iv), we have that q‘dp. The last statement of (vi) follows from‘Theorem 2.25. U Corollary 2.27: Let G be a finite group having all of its prOper normal subgroups p-nilpotent and ‘G‘ I pam where (p,m) I 1 and a E 1. Suppose also that either condition (a) or (b) of Theorem 2.26 holds. Then G is p-nilpotent if and only if there exists a proper normal subgroup K of G with (‘G:K‘,dp) I 1, where dp I 3 (pi - 1). i=1 Eppgf: If G is p-nilpotent, then there exists K‘g G with ‘GzK‘ I p8 which is prime to dp. On the other hand, if G is not p-nilpotent, then by (iv) of the preceding theorem, every prOper normal subgroup of G has index prime to dp. H In Theorem 2.26 and its corollary, we have required that all the proper normal subgroups of G be p-nilpotent. If we now extend this requirement of p-nilpotence to the larger class of somewhat normal subgroups of G, as defined below, we recover all of Theorem 2.14. Definition 2.12: Let H be a proper subgroup of a given finite group G. H will be said to be somewhat normal in G if cofGH I H/corGH is cyclic of prime-power order. 41 Theorem 2.28: Let G be a finite non-p-nilpotent group having all of its proper somewhat normal subgroups p-nilpotent. Suppose also that one of the following two conditions is satisfied: (a) The subcofactor of each proper self-normalizing subgroup of G is p-nilpotent. (b) The subcofactor of each proper abnormal subgroup of G is p-nilpotent and either p is odd or the p-Sylow subgroups of G are abelian. Then: (i) ‘G‘ I paqb for some prime q I p; in particular, G is solvable. (ii) All proper subgroups of G are nilpotent. (iii) The conclusions of Theorems 2.14 and 2.26 hold. Egggf: (i) It follows from‘Theorem 2.26 that G has a normal p-Sylow subgroup P. We consider first the case where P is not a min- imal normal subgroup of G. Then there exists M14 G with I'M i P. (ii/M) I scof Since scof H for each prOper subgroup H/M of G/M (by Chi G Lemma 1.4), and since H is self-normalizing (abnormal) in G if H/M is self-normalizing (abnormal) in G/M, it follows that hypothesis (a) or (b) holds in G/M according as (a) or (b) holds in G. Also, the proper normal subgroups of G/M are clearly p-nilpotent. Finally, G/M is not p-nilpotent. For if T/M were a normal p-complement of G/M, then T would be a prOper normal subgroup of G since p“G/M‘, and hence p- nilpotent by hypothesis, say with normal p-complement U. As in the proof of the preceding theorems, it then follows that U would be a normal p-complement of G; but this contradicts the non-p-nilpotence of G. 42 The hypotheses are thus satisfied by G/M. Since p“G/M‘, we have by induction, that ‘G/M‘ I pkqb for some prime q I p. Therefore, b - ‘G‘ I paq where ‘M‘ I p8 k. Now consider the case where P is minimal normal in G and thus r b, elementary abelian. Let ‘G‘ I pa fl qi:1 where p and the qi are dis- i=1 tinct primes dividing ‘6‘. Suppose now that r > 1. Then for each i I l, ...,r, if Qi is any qi-Sylow subgroup of G, P01 is a proper subgroup of G, and thus so also is P<:x:> for each x E Qi' Now, since P 4 G, we have P C corG(P) so that P/corG(P) is a homo- morphic image of P<:x:>/P E‘<}(> and is therefore a cyclic qi-group. Hence, for each i I 1,... ,r and for each x E Qi’ P<:x:> is a somewhat normal subgroup of G and, consequently, is p-nilpotent by hypothesis. This implies that is normal in P so that x centralizes P; and since this is true for each x E Qi’ we have Qi C CG(P) for each i I 1,... ,r. Since P is abelian, we also have P C CG(P). It follows that G I C CG(P), that is, P C Z(G). But by (2) of Theorem 2.18, this implies that G is p-nilpotent, a contradiction. b Therefore, r I 1, and ‘G‘ I paq1 1. The solvability of G now follows from the well-known theorem of Burnside that groups of order paqb, where p and q are primes, are solvable. (ii) Since P14 G and ‘G‘ I paqb, G is q-nilpotent so that all Subgroups of G also are q-nilpotent. Let K be a maximal normal subgroup of G containing P. Then K is q-nilpotent, and by hypothesis, K is p- nilpotent; consequently, K is nilpotent. Since G is solvable, G/K is of prime order; and since P C K, we have ‘G/K‘ I q. Now let S be any maximal subgroup of G; by what we 43 have shown, S is q-nilpotent. Either S I K, in which case S is nil- potent, or SK I G. In this latter case we have ‘8 n K‘ I SG K I 13L . By the nilpotence of K, S n K 4 4 K4 G, that is, S n K 4 <1 G, and thus S n K C scorGS I corGS. Hence, S/corGS has order 1 or q so that S is somewhat normal (in fact, nearly normal) in G. By hypothesis, there- fore, S is p-nilpotent; and since S is also q-nilpotent, this means that S is nilpotent. Thus all the maximal subgroups (and hence all proper subgroups) of G are nilpotent. (iii) now follows immediately from (ii). 0 We conclude this section with the following result in which, as in the preceding theorem, we again strengthen the conditions imposed in Theorem 2.26. Since the p-nilpotence of a subgroup or of the subcofactor of a subgroup provides no useful information in the case that this subgroup or subcofactor has order prime to p, we would hope to obtain more of the structure of G if we impose some additional condition on these. Although, by Example 2.13, we cannot hope to recover all of Theorem 2.14 under the conditions imposed in the fol- lowing theorem, we do, nevertheless, obtain some additional information about C. To state the result we need the following definition. Definition 2.13: A finite group G will be said to be (ng)- ilpotent if: (i) G is p-nilpotent; (ii) q“G‘ and G is q-nilpotent in case p Y‘G‘ and ‘G‘ > 1. Theorem 2.29: Let G be a finite group with p a prime factor of ‘G‘ for which every proper normal subgroup K is (p:qK)-ni1potent for some prime qK depending on K. Suppose also that the cofactor H/cor H of each proper subgroup H of G is (p:qH)-nilpotent for G some prime qH depending on H. Then the following hold: 44 (i) G is solvable. (ii) G has a normal Sylow subgroup. (iii) If G is not p-nilpotent, then the conclusions of Theorem 2.26 hold; in particular, G has a normal p-Sylow subgroup P C G'; Fp(G) is the unique maximal normal subgroup of G and ‘G/FP(G)‘ I q for some prime q I p; if P is abelian, then G has exactly pa distinct p-complements where ‘G‘ I pam with (p,m) I l. ngpfi: (iii) follows immediately from Lemma 1.2, part (i), and Theorem 2.26; only (i) and (ii) require proof. For these, we consider two cases. r a. Case 1: G is p-nilpotentu-Let ‘G‘ I .2 pi1 where p1 I p and the pi are distinct primes dividing ‘G‘. We miylassume that r > 1, since the result is trivially true otherwise. Then there exists a T 4 G with ‘Gle‘ I plpl since G is pl-nilpotent. Since T1 I l is a proper normal pi-subgroup of G, it is, by hypothesis, pi-nilpotent for some i I 2, say for i I 2. Thus there exists T2 characteristic in T1, hence normal in G, with ‘T1:T2‘ I p:12 . Continuing gives a normal series of G, (}==TO J'T1.>T2.J ...:)Tr_1:51}.Iv’l‘3 where for each i I 1, 2, ..., r, ‘Ti-l/Ti‘ I p:li . It follows that G is solvable with a normal pr-Sylow subgroup Tr-l' Case 2: G is not p-nilpotent.-Then by Theorem 2.26, G has a normal p-Sylow subgroup P so that (ii) holds. We consider separately the two possibilities that P is or is not a minimal normal subgroup of G. (a) Suppose P is not minimal normal in G. Then there exists M14 G with < l>'f M‘g P. We show that the hypotheses are satisfied 45 by G/M. If H/M is any prOper subgroup of G/M, then H is a prOper sub- group of G. By hypothesis, cofGH I H/corGH is (p:qH)-ni1potent for some prime qH depending on H. From Lemma 1.4, we have that cof G/Mm/M) is isomorphic to cof H and is therefore (p:qH)—nilpotent relative to G this same prime qH. Clearly, all proper normal subgroups of G/M are p-nilpotent. Now suppose K/M #' is a pr0per normal p'-subgroup of G/M. Then K32 G, hence is p-nilpotent by hypothesis, say with normal p-complement T. Since T is characteristic in K, T is normal in G; and since M is the p-Sylow subgroup of K, K I M X T. Now since T is a nontrivial proper normal p'-subgroup of C, it is q-nilpotent for some prime q dividing ‘T‘. Hence, since K/M I T, q divides ‘K/M‘ and K/M is q-nilpotent. The hypotheses thus hold for G/M so that, by induction, G/M is solvable. And since the p-group M is solvable, it follows that C also is solvable. (b) Suppose now that P is minimal normal in G. Then either there exists a minimal normal subgroup L of G which is distinct from P, or else P is the unique minimal normal Subgroup of G. In the first case, we have 1.0 P I- so that L is a prOper normal p'-subgroup of G, hence is q-nilpotent for some prime q dividing ‘L‘. Since the normal q-complement of L is characteristic in L, it follows from the minimality of L that L is a q-group. As in (a), the conditions on the cofactors of subgroups of G/L are satisfied, and all proper normal subgroups of G/L are p-nilpotent. Now let K/L # < 1) be a proper normal p'-subgroup of G/L. Then Since L 46 is a p'-group, K is a proper normal p'-Subgroup of G, and hence is, by hypothesis, ql-nilpotent for some prime q1 dividing ‘K‘. Let U be the normal ql-complement of K. Now, if ql“K/L‘, we have exhibited a prime q1 dividing ‘K/L‘ for which K/L is ql-nilpotent. Suppose, there- fore, that ql does not divide ‘K/L‘. This means that q1 I q, from which it follows that K I U X L. Now since U is a nontrivial proper normal p'-subgroup of C, it is qz-nilpotent for some prime q2 dividing ‘U‘. Since K/L I U, we have that qz“K/L‘ and that K/L is qZ-nilpotent. The hypotheses thus hold for G/L so that, by induction, G/L is solvable. Since the q-group L is solvable, this means that G also is solvable. There remains to consider only the possibility that P is the unique minimal normal subgroup of G. By the Schur-Zassenhaus Theorem, P has a complement T in C. Now if H is any subgroup of T (not neces- sarily proper), H is a proper p'-subgroup of G; and since it does not contain the unique minimal normal subgroup P of G, we have corGH I so that H I cofGH. By hypothesis, therefore, each nontrivial subgroup H of T (including T itself) is qH-nilpotent for some prime qH dividing ‘H‘ (where qH depends on H). t b Let ‘G‘ a pa U qi 1, where p and the qi are distinct primes i=1 t b, dividing ‘G‘. Then T, being a p-complement of G, has order H qil'. i=1 Since T is qi-nilpotent for some i, say for i I 1, there exists b _ l 4 T with ‘Tle‘ - q1 T T1 is qi-nilpotent for some i E 2, say for l i I 2, and thus there exists T2 characteristic in T1, hence normal in b T, such that ‘leTz‘ I q222. Continuing gives a normal series of T, T T DTlDT D...3Tt_1:>Tt=<1>, 0 2 b qii' for each i I l, 2, ..., t. Therefore, T is with ‘Ti_1/Ti‘ 47 solvable. Since G/P I T, we have G/P solvable; and since the p-group P is solvable, it follows that C also is solvable. U 2.4 Sylow-towered and Supersolvable Cofactors or Subcofactors In this section we examine the influence on a finite group G of supersolvable subcofactors of certain subgroups of G and, more generally, of o-Sylow-towered subcofactors where o is some fixed ordering of a set 2 of primes containing n(G) I the set of prime factors of ‘G‘. Our goal is to extend the well-known theorem of Huppert__ If all the proper subgroups of a finite group G are Supersolvable, then G is solvab1e__ and some extensions of this result by Rose [16, 17], who required that only the self-normalizing or abnormal subgroups of G be supersolvable, or, more generally, o-Sylow-towered. The concept of a group being c-Sylow-towered is defined as follows. Definition 2.14: Let G be a given finite group and n(G) the set of prime factors of ‘G‘. Let G I (p1, p , pt) be a fixed 2,... ordering of a set 2 of primes containing n(G). Then G is said to have a g:Sylow-tower if there exists a normal series (11) I GO C G1 C ... C CC I G such that for each i I 1, 2,... , t, Gi/Gi is isomorphic to a pi-Sylow subgroup -l of G (which we allow to be trivial in case pi I ‘6‘). For example, let 2 I {p1, p2, ..., pm] 2 n(G) and o the natural descending order of 2, say 0 I (p1, p2, ..., pn) where p1 > p2 >»...>'pn. It is well-known (see, for example, M. Hall [9], 10.5.3) that if G is supersolvable, then G has a o-Sylow tower for this a. 48 We might mention that Doerk in [6] has extended the theorem of Huppert, which was stated above, by describing much of the Structure of G. Several of his results parallel those of the Schmidt-Iwasawa Theorem (Theorem 2.3); some of these are given in the following theorem. Theorem 2.30: Let G be a finite group all of whose proper subgroups are supersolvable. Then: (i) G is solvable; (ii) G has a o-Sylow tower where o is the natural descending order of n(G), or G is a nonnilpotent group having all its proper subgroups nilpotent. If G itself is not Supersolvable, then the following also are true: (iii) G has exactly one normal Sylow subgroup P. (iv) §(P) C Z(G) so that cl(P) § 2; exp(P) I p for p odd and exp(P) S 4 for p I 2, where P is a p-group; Q(P) is super- solvably embedded in G, that is, there exist normal sub- groups N1 of G such that IN0 C'. N C CZNm I §(P) l and ‘Ni/N I p for each i I l, ... , m. 1.11 (v) ‘G‘ is divisible by at most three distinct primes. Our first result in this direction follows from Corollary 2.2 of Section 1. Before stating it, however, we first establish the following lemma. Lemma 2.31: For a given group G, F2(G') I G', that is, G' has Fitting length E 2, if and only if G/F2(G) is abelian. Proof: We show first that F(G') I F(G) n G'. For this, we have that since F(G') is characteristic in G' which is normal in G, F(G') is 49 normal in G and is nilpotent, hence is contained in F(G); consequently, F(G') C F(G) n G'. But F(G) n G' is normal in G' and is nilpotent, hence is contained in F(G'). Thus, F(G') I F(G) n G'. Using this equality and the normality of G' and F(G), we now have the following chain of equivalent statements: F2(G') = G' if-f G'/F(G') is nilpotent ifff G'/F(G) n G' is nilpotent if-f G'F(G)/F(G) is nilpotent if—f G'F(G)/F(G) : F(G/F(G)) = F2(G)/F(G) if-f G'F(G) c. F2(G) if-f G' c F2(G) if-f G/F2(G) is abelian. [1 Theorem 2.32: Let G be a finite solvable group for which the cofactors of all maximal subgroups are supersolvable. Then: (i) clue) is supersolvable. (ii) §(G') E 2; that is, F2(G') I G', or equivalently, G/F2(G) is abelian. (iii) j-‘(m a 3; that is, F3(G) = G. 2593;: (i) Supersolvability is clearly a strictly homomorphism- invariant property in the sense of Definition 2.2. Thus, since G is solvable and cofactors of maximal subgroups are supersolvable, it follows from Corollary 2.2-(ii) that G/F(G) is supersolvable. (ii) Since G/F(G) is supersolvable, its derived subgroup (G/F(G))' is nilpotent. By the semark (l) preceding Lemma 2.8, (G/F(G))' = G'F(G)/F(G), which is isomorphic to G'lc' n F(G), and which is in turn equal to G'/F(G') by the proof of Lemma 2.31. There- fore, G'/F(G') is nilpotent so that F2(G') I G'. 50 (iii) From Lemma 2.31, it now follows that G/F2(G) is abelian, hence nilpotent, and thus G has Fitting length E 3. fl That m I 2 and n I 3 are the best possible integers for which G' I Fmic') and G I Fn(C) in the preceding theorem is Shown by the following example. Example 2.33: Let S4 be the symmetric group on 4 letters. Then S4 is solvable, the cofactors of all proper subgroups of 84 are supersolvable, the Fitting length of SA I A4 (the alternating group of degree 4) is 2, and the Fitting length of S4 is 3. Proof: The solvability of S is well-known. The Fitting sub- 4 group of SA I A4 is the four-group V, from‘which it is clear that f(A4) I 2. Also, it is immediate that F(S4) I F1(S4) I V and F2(S4) I A4 so that “34) = 3. 4 having order 4 or 12 are normal in S4 and hence have trivial cofactor. The subgroups of order 1, 2, or 3 are The subgroups of S obviously supersolvable, and thus so also are their cofactors. The only other subgroups of 84 that need be checked are those of order 6. These have trivial core and are isomorphic to the group 83 which is supersolvable. Consequently, the cofactors of all prOper subgroups of 84 are supersolvable. n In Example 2.17 of the preceding section, a group G was con- structed which was not solvable, but in which all the proper abnormal subgroups were supersolvable. This shows that the hypothesis of G being solvable in Theorem 2.32 cannot be omitted; that is, the super- solvability of the cofactors of all maximal subgroups of G is not sufficient to guarantee that G is solvable. As we now Show, however, if we enlarge the class of subgroups which are to have supersolvable 51 cofactors or subcofactors (more generally, o-Sylow-towered subco- factors) from the nonnormal maximal subgroups to the collection of all self-normalizing subgroups of G, then G is solvable. This extends the following result due to Rose [16]. Theorem 2.34: Let a be a fixed ordering of a set 2 of primes containing n(G). If every prOper self-normalizing subgroup H of G has a o-Sylow tower, then G is solvable. Theorem 2.35: Let a be a fixed ordering of a set 2 of primes containing n(G). If the subcofactor H/scorGH of each self-normalizing sub- group H of G has a o-Sylow tower, then G is solvable. Moreover, G/F(G) has a O-Sylow tower. 2322;; 'We first establish the solvability of G, using induction on ‘G‘. If G is simple, then every proper subgroup has subnormal core Ii<21>, and hence has a o-Sylow tower. The solvability of G then follows by Theorem 2.34. Thus, suppose that G is not simple, and let M.be a minimal normal subgroup of G. If H'I H/M is any proper self-normalizing subgroup of G'I G/M, then clearly H-is a proper self-normalizing subgroup of C. By hypothesis, H/scor H is a-Sylow-towered; hence, from Lemma 1.4, so G also is H/scorca'i) I H/scor H. Thus, since n(G/M) C 11(6) C )3, the G hypotheses hold for G'I G/M so that, by induction, G/M is solvable. We Show now that the hypotheses hold for M. 'For this, let H be any self-normalizing (in M) prOper subgroup of M. Then.N I N;°(H) I the hypernormalizer of H in G is a self-normalizing subgroup of C. Now, M.¢ N; for otherwise, since H‘4'4'N, we would have H 414 M, in contradiction to the fact that H is self-normalizing in M. Thus, N f G; by hypothesis, therefore, N/scoréN has a o-Sylow tower and 52 hence, so also does its subgroup HscorGN/ scorGN. Now, since H14 4 N, we have H n scorGN‘4‘4 scorGN 4 4 C so that H n scordN is subnormal in G and thus must be contained in scorGH. Hence, H n scor N C scor H C scorMH (the last inclusion being true G G since scorGH subnormal in G implies that it is subnormal in M). Since H/H n scorGN, being isomorphic to HscorGN/ scorGN, has a c-Sylow tower, so also does its homomorphic image H/scorMH. Thus, since n(M) C n(G) C 2, the hypotheses hold for M,so that, by induction, M is solvable. And since G/M is solvable, it follows that C also is solvable. To show that G/F(G) has a o-Sylow tower, we need only show that the property T0 of having a o-Sylow tower is a strictly homomorphism- invariant property in the sense of Definition 2.2. For the hypotheses of the theorem imply that S/scorGS I S/corGS I cofGS is a Ta-group for all maximal subgroups S of G, since each maximal subgroup of G is either normal, and thus has trivial cofactor, or is self-normalizing in G. Also, we have proved that G is solvable. Therefore, if To is a strictly homomorphism-invariant property, then by Corollary 2.2-(ii), G/F(G) is a Ta-group. The fact that To is strictly homomorphism-invariant is almost immediate. For let < 1>" G C G C ... C Gt I G be a o-Sylow tower 0 l of G. If ¢ is a homomorphism of G onto 5, then clearly <1> = ¢(GO) C ¢(G1)C C ¢(Gt) I G is a c-Sylow tower for G; consequently, To is homomorphism-invariant. Also, if H is a subgroup of G, then <1>=anocnnclc...chnct=H 53 is a o-Sylow tower for H; thus, To is subgroup-inherited. Finally, if (11) I Ko C K1 C ... C Kt I K is a c-Sylow tower of the group K, then <1>IGOXKOCGIXK1C...CGtXKt=GXK is a o-Sylow tower for G X K. Therefore, To is a strictly homomorphism- invariant prOperty, and the result now follows. B We have already commented that a supersolvable group G has a o-Sylow tower for o the natural descending order of a set 2 of primes containing n(G). The following corollary now is an immediate conse- quence of the preceding theorem and Theorem 2.32. Corollagy 2.36: If the subcofactor H/scorGH of each proper self- normalizing subgroup H of G is supersolvable, then G is solvable. Moreover, G/F(G) is supersolvable and G/F2(G) is abelian; thus, f(c') a 2, §(c) a 3. It might be mentioned that Rose in [17] has shown that, in com- parison with Doerk's result (Theorem 2.30), for all n > 1, there exists a group G such that ‘n(G)‘ I n, G is not supersolvable, but every self- normalizing proper subgroup of G is cyclic. Thus, assuming that G is not Supersolvable, or more generally, not o-Sylow-towered, in these results imposes no bounds on the number of prime factors of ‘G‘. AS we have already seen, the group G in Example 2.17, constructed by extending the simple group of order 168 by an automorphism of order 2, is not solvable but has every proper abnormal subgroup supersolvable. One cannot, therefore, replace "self-normalizing" by "abnormal" in the three preceding results without imposing some additional condition. Rose in [16] has shown, however, that the following is true. Theorem 2.37: Let a be a fixed ordering of a set E of primes containing n(G). If every prOper abnormal subgroup H of G has a o-Sylow 54 tower and the 2-Sylow subgroups of G are abelian, then G is solvable. Theorem 2;3§: Let c I (p1, p2,... , Pn) be a fixed ordering of the set {p1, p2,... , pn] of primes containing n(G). If the subcofactor H/scorGH of each proper abnormal subgroup H of G has a O-Sylow tower and the 2-Sylow subgroups of G are abelian, then G is solvable. Mereover, G/F(G) has a 0-Sylow tower. 2523;: The last statement follows as in the proof of Theorem 2.35 so that only the solvability of G requires proof. For this, we suppose that it is not solvable, and let G be a minimal counterexample. Then G has no proper nontrivial solvable normal subgroup. For if K were such a subgroup, then by the minimality of G, G/K would be solv- able, since it is easily checked that the hypotheses hold for G/K. Since K is solvable, this would mean that G is solvable. Thus, G has no nontrivial normal solvable subgroup and hence, from Lemma 2.19, no nontrivial subnormal solvable subgroup. Now define the integer r é n by the following conditions: There exists a normal chain Gr C Gr C ... C Gn I G such that Gr is not +1 pr-nilpotent; and in case r < n, Gi/Gi is isomorphic to a pi-Sylow -1 subgroup of G for each i I r + 1,... , n. Since G is not solvable and groups having a 0-Sylow tower are solvable, we have r > 0, so that H I Gr *‘< l>. There are two possibilities: (1) pr I 2, and (2) pr is odd. Case 1: pr I 2.-Let P be a 2-Sylow subgroup of H I Gr' Then, since 2 does not divide ‘G:H‘, P is a 2-Sylow subgroup of G. Thus, N I NG(P) is abnormal in G, and N f G since G has no nontrivial normal solvable subgroups; by hypothesis, therefore, N/scorGN has a o-Sylow 55 tower. Now, since P-4 N, we have P n scorGN 4 scorGN 4.4 C so that I’m scorGN 4‘4 G. Since G has no nontrivial subnormal solvable sub- groups, Pr) scordN I <;1>3 that is, scorGN is of odd order, hence is solvable by the Feit-Thompson Theorem. Again, since scorGN 4‘4 G and G has no nontrivial subnormal solvable subgroups, we have scorGN I < l>. Therefore, N I N/scorGN has a o-Sylow tower, and thus so also does its subgroup NH(P) I N n H. Now, since pr+1"'° , pn do not divide ‘H‘, it follows that NH(P) has a (p1,... , pr)-Sylow tower; in particular, NH(P) is pr- nilpotent, say with normal pr-complement (I 2-complement) T. Since both P and T are normal in NH(P)’ we have NH(P) I P X T; consequently, T centralizes P. But P is abelian by hypothesis so that P C CG(P). It now follows that P C Z(NH(P)), and hence, by Theorem 2.18, H I Gr is pr-nilpotent. This, however, contradicts the choice of r. Case 2: pr is oddr——Let P be a pr-Sylow subgroup of H I Cr and thus, as in Case 1, a pr-Sylow subgroup of C also. Let J(P) be the Thompson subgroup of P, as defined in Def. 2.11, and let N I NG(Z(J(P))). Now NG(P) is abnormal in G; and Z(J(P)) char é P14 NG(P) implies that Z(J(P)) is normal in NG(P) so that NG(P) C N. Consequently, N also is abnormal in G. And since G has no nontrivial normal solvable sub- groups, we have N I C. By hypothesis, therefore, N/scor N has a o- G Sylow tower, and hence so also does its subgroup (NH)(scorG§)/scorc§ where NH I NH(Z (J (P))) I N n H. It follows that NH/ NH 0 scorGN, being G§)/scorcfi, also has a o-Sylow tower. Now, pr+l’ ..., pn do not divide ‘H‘ and thus do not divide ‘NH‘. g isomorphic to (NH)(scor Consequently, NH/NIi n scorGN has a (p1,... , pr)-Sylow tower. In particular, NH/ NH n scorGN is pr-nilpotent. 56 ~ ~ Suppose now that P I P n scorcfi Ii. Then since scorGN 4 N, 1 we have I P1'4 P so that P1 must intersect Z(P) nontrivially. Since Z(P) C Z(J(P)) by the note following Definition 2.11, this implies that P2 I P1 0 Z(J(P)) is also nontrivial. Therefore, .<:l> I P2 = P1 0 Z(J(P)) = P n scorc§ n Z(J(P)) = Z(J(P)) n Scorcfi. But since Z(J(P)) 4 N, we have <11) I P2 I Z(J(P)) n scorGN 4 scorGN 4‘4 C so that I P2'4 4 G; and this contradicts G having no nontrivial subnormal solvable subgroups. Thus Pr] scordN I so that scorGN is a pé-group, and hence a a :3 z z so also is NH n scorGN. But since NH/li n scorCN is pr-nilpotent, this H implies that N I NH(Z(J(P))) is also pr-nilpotent. By the Glaubermann- H Thompson Theorem CThm. 2,22), it follows that H I Gr is pr-nilpotent, which again contradicts the choice of r. Each case, therefore, leads to a contradiction; and we conclude that no such minimal counterexample can exist. fl From this theorem and Theorem 2.32, we have the following corollary. Corollary 2.39: If the subcofactor H/scorGH of each proper abnormal subgroup H of G is supersolvable and the 2-Sylow subgroups of G are abelian, then G is solvable. Moreover, G/F(G) is supersolvable and G/F2(G) is abelian; thus, §(G') E 2 and fm)53. Rose has also established in [17] the following result: Thoerem 2.40: If every proper abnormal subgroup H of G is super- solvable and the abnormal maximal subgroups have prime-power index, then G is solvable. 57 Here again it is sufficient to require only that the subco- factors of the abnormal subgroups of G be supersolvable. To establish this, we need the following lemmas, the first of which is due to GaschUtz [7]. Lemma 2.41: For a given group G, let F(G) be the intersection of all the abnormal maximal subgroups of G, and §(G) the Frattini subgroup of G. Then F(G)/§(G) I Z(G/§(G)); in particular, F(G) is a normal nilpotent subgroup of G. Lemma 2.42: Suppose that G is a simple group and that G I HK where H and K are pr0per subgroups of G. Then corH(H G K) I corKO-I n K) I < l>. .Egggg: Let C I corH(H n K). Then, -1 CGI < (hk)-lc(hk) ‘h e H, k e K> < k-ICk‘k E K> (since C 4 H) so that CG C K‘i G. Thus, CG is a prOper normal subgroup of G. Since G is simple, we have CG I a Therefore, C I < 1>»also. Similarly, corKO-I n K) I < l>. [1 Theorem 2.43: If the subcofactor H/scorGH of each prOper abnormal subgroup H of G is supersolvable and the abnormal maximal subgroups of C have prime-power index, then G is solvable. Moreover, G/F(G) is supersolvable and G/F2(G) is abelian; thus, f(G') E 2 and f(G) E 3. 2523;: The last part follows as before, and only the solv- ability of G requires proof. For this, we proceed by induction on ‘G‘. If G is simple, then all maximal subgroups of G are abnormal in G and have subnormal core 1, hence are supersolvable by hypothesis, 58 so that G is solvable by Theorem 2.30 or the preceding theorem. Thus, suppose that G is not simple, and let M be a minimal normal subgroup of G. Now, if H'I H/M is any prOper abnormal subgroup of GiI G/M, then clearly H is a prOper abnormal subgroup of G; by hypothesis, H/scor H is supersolvable, and hence, by Lemma 1.4, so G also is H73cor§(fi) I H/scorGH. And if §.= S/M is any abnormal max- imal subgroup of G, then S is an abnormal maximal subgroup of G so that ‘GtS‘ I ‘G:S‘ I a power of a prime. The hypotheses therefore hold for G/M; consequently, G/M is solvable by induction. If G has another minimal normal subgroup MI I M, then G/M* is likewise solvable by induction, hence so also is G/M X G/M*. And since G I G/M n M? is isomorphically embedded in G/M X G/MI, it follows that G also is solvable. We may assume, therefore, that M is the unique minimal normal subgroup of G, and we need only Show that M is solvable. We assume that it is not, and will Show that this leads to a contradiction. So suppose that M is not solvable. Then M I M1 X ... X Mk’ where the Mi are isomorphic simple nonabelian groups. From Lemma 2.41, F(G) I the intersection of all abnormal maximal subgroups of G is a normal nilpotent subgroup of G. Since M is not solvable, we have M,¢ F(G), from'which it follows that there exists an abnormal maximal Subgroup S of G not containing M. By the uniqueness of M, we have scor S I cor G GS =., and thus, G possesses maximal subgroups of core 1. Suppose now that there exists only one conjugacy class C of maximal subgroups of core 1. Then, by hypothesis, for some power p8 of a prime p, ‘G:S‘ I p8 for all S 6 Ca Since M.¢ S for S E C, we 59 have MS I C so that pa I ‘G:S‘ I ‘MS:S‘ I ‘M:M n S‘, and hence p“M‘. Let P be a p-Sylow subgroup of M; then, I P's M since M is not solvable and p“M‘ . Now, from the Frattini argument, G I FNG(P); and by the minimality of M, NG(P) I G. Thus NG(P) C T for some maxi- mal subgroup T of G. Since G I MNG(P) I MT, we have M i T, hence corGT I < 1) by the uniqueness of M. It follows that T E C, so that p8 I ‘G:T‘ I ‘MTzT‘ I ‘M:M n T‘. This, however, contradicts the fact that since P ; M10 NG(P) c M.n T, py‘Mzm n T‘. Therefore, 6 has at least two distinct conjugacy classes of maximal subgroups of core 1. Now let S be a maximal subgroup of G with cor S I scorGS I < 1), G and let P be a p-Sylow Subgroup of S, where p I max(n(S)). Since 3.4 c, s is abnormal in c, and hence, S = S/scor S is supersolvable G by hypothesis. This means, in particular, that since p is the great- est prime factor of ‘S‘, P is normal in S so that S C NG(P). And since G has no nontrivial solvable normal subgroups, it follows from the maximality of S that S I NG(P); also, it is now clear that P is a p-Sylow subgroup of G. Thus, if S and T are maximal subgroups of G with core 1 and P = max(n(S)) I max(n(T)), then S I NG(P) and T I NG(P*) for some p-Sylow subgroups P and P* of G. Since P and P* are conjugate in C, so also are S and T. Therefore, if S is any maximal subgroup of G with core 1 and p I max(rr(S)), then the conjugacy class of S I 0(1)) I {NG(P)‘P a p-Sylow subgroup of G]. Let now C(pl) and C(pz) be two such conjugacy classes of maxi- mal subgroups of core 1, with p1 > p2. Then p1 I p I the greatest prime factor of ‘G‘. For if T E C(pz), then p2 I max(n(T)); and since p 2 p1 > p2, this implies that both p and p1 divide ‘G:T‘. By 60 hypothesis, ‘GzT‘ is a power of a prime, and thus p I p1. Let S E C(pl) I 64p) and T G C(pz). Then ‘G:T‘ I power of p and pI‘T‘ so that T is a p-complement of G. Let q“G:S‘ and U be a q-complement of S, hence of G. (U exists since S'I S/scorGS is super- solvable, hence solvable.) Then, since M‘4 G, T n M and U n M are p- and q-complements respectively of M; and since M is a direct product of the Mi’ T 0 Mi and U n Mi are p- and q-complements respectively of Mi’ for each i I 1,... , k. From this it follows that Mi I (T n Mi)(U 0 Mi); and since T n Mi and U n M,i are supersolvable while M is not, these must be pr0per subgroups of Mi for each i. By Lemma 2.42, therefore, Ai I T n U n M.i contains no nontrivial normal subgroup of either T 0 Mi or U 0 “1' Now, A1 is a Hall {p ,q]'-subgroup of M1 for each i; for p and q do not divide ‘Ai‘ , and ‘Ai‘ = ‘T n Mi“U n Mi‘l‘Mi‘ so that ‘Mi:Ail I ‘Mi:T n MillMi:U 0 Mi‘ I paqb for some a and b. It follows that Ai is a q-complement of T 0 M1 and a p-complement of U 0 M1. We assert now that A1 is abelian. For this, let P be a p-Sylow subgroup of U 0 Mi' Since p I max(n(U 0 Mi)) and U 0 Mi is super- solvable, we have P.4 U 0 Mi so that P C F(U 0 Mi)’ the Fitting sub- group of U n Mi' If P were properly contained in F(U 0 Mi)’ then F(U n Mi) would have a nontrivial normal p-complement K which, being . But then K must characteristic in F(U 0 Mi)’ is then normal in U 0 Mi be contained in the p-complement A1 of U 0 Mi’ which contradicts the fact that Ai contains no nontrivial normal subgroup of U 0 Mi' Thus, P I F(U n Mi); and since U 0 M1 is supersolvable, (U n Mi). C F(U 0 Mi) I P so that (U n Mi)/P‘I A1 is abelian. 61 Also, we have q I max(n(Tl] Mi))' For let q' be the greatest prime factor of ‘T n Mi" and let Q be a q'-Sylow subgroup of T 0 Mi; since T 0 Mi is supersolvable, Q is normal in T n Mi' And since Ai contains no nontrivial normal subgroup of T 0 Mi’ Q C Ai; hence q"‘T n MizAi‘, which means that q' I q. From this fact that q I max(n(T n Mi)), it follows, in particular, that q I min(n(Mi)). For if it were, then T 0 Mi would be a q-group, making Mi a {p1,q]- group and hence solvable by Burnside's Theorem. Now let A I T n U n M; then A is a Hall {p ,q]'-subgroup of M, and A is abelian as the direct product of the abelian groups Ai. Let r I min(n(M)); by what we have just shown, p > q > r. Let R be an r-Sylow subgroup of A, hence of M; R is abelian. We claim that N I NG(R) is a proper abnormal subgroup of G. That N I G is clear, since G contains no nontrivial solvable normal subgroups. Now, since M‘4 G, R I RI n M for some r-Sylow subgroup R? of G. If x 6 NG(R*), then Rx = 01*)" n M" = 11* n M = R, thus x e NG(R). This shows then that NG(R*) C NG(R); and since NG(R*) )4 G, NG(R) also is abnormal in C. By hypothesis, therefore, N/scordN is Supersolvable, and thus I N (R) I‘M n N. M M Hence, NM/NMrfl scoréN I (NM)(scorGN)/scordN lS supersolvable; and since r I min(n(M)),‘NM/NM|1 scorGN is r-nilpotent, say with normal so also is its subgroup (NM)(scorGN)/scor¢N where N r-complement L/NM.n ScordN. Now, since R is normal in N I NG(R), we have R.n scor N14 scor N; G G and since scordN‘4‘4 C, it follows that R n scordN 4‘4 G. Because G has no nontrivial solvable normal subgroups and hence, by Lemma 2.19, no nontrivial solvable subnormal subgroups, we have R.n scordN I < l>. 62 This means then that NM 0 scorCN is an r'-group, and thus that L is a normal r-complement of N I NM(R). It follows that NM(R) I R X L M so that L centralizes R. And since R is abelian, R C CG(R); hence R,C Z(NM(R)). But, by Theorem 2.18, this implies that M is r- nilpotent, and thus has a proper characteristic subgroup, which contradicts the minimality of M. Therefore, M is solvable, and the result now follows. Q As an addition to Corollaries 2.36, 2.39, and the preceding theorem, we have the following result. Theorem 2.44: Let G be a finite nonsupersolvable group having all proper normal subgroups supersolvable. Suppose also that one of the following two conditions holds: (a) The subcofactors of the proper self-normalizing subgroups of G are supersolvable. (b) The subcofactors of the prOper abnormal subgroups of G are supersolvable and either the 2-Sylow subgroups of G are abelian or the abnormal maximal subgroups of C have prime- power index . Then the conclusions of Theorem 2.43 hold, and G has a normal p-Sylow subgroup for p the least or largest prime factor of ‘G‘. [23922: Let p I min(n(G)). Then the proper normal subgroups and the subcofactors of the self-normalizing (abnormal) subgroups of G are p-nilpotent. If G is not p-nilpotent, then by Theorem 2.20 or 2.23, G has a normal p-Sylow subgroup. On the other hand, if G is p-nilpotent, say with normal p-complement T, then T is, by hypothesis, supersolvable. Thus if q is the largest prime dividing ‘G‘, hence the greatest prime 63 factor of ‘T‘, T has a normal q-Sylow subgroup Q. Then Q, being characteristic in T, is normal in G; and since ‘GzT‘ is prime to q, Q is a normal q-Sylow subgroup of G. U CHAPTER THREE THE INFIUENCE ON A GROUP OF THE OUTER COFACTORS OF ITS SUBGROUPS 3.1 Introduction; Definitions and Basic Properties For H a proper subgroup of a given finite group G, corGH was defined as the maximal G-normal subgroup contained in H. In the pre- ceding chapter we have considered the effect on G of conditions imposed on corGH and cofGH I H/corGH (or scofGH), where H ranges over a certain class of proper subgroups of G. Now, one might hope to be able to "dualize" some of the results obtained. Thus, we consider for a given prOper subgroup H of G, those subgroups which are outside H, or at least not contained in H, and which are in some sense minimal with reSpect to the normal structure of G. Following basically the ideas suggested by Deskins in [5], we make the following definitions. Definition 3.1: For H a proper subgroup of a finite group G we let OG(H) denote the collection of subgroups C of G which satisfy the following conditions: (i) CC H; (ii) each proper G-normal subgroup of C is contained in H. Notice that if C €<2C(H) and C‘4 G, then C is minimal with respect to being normal in G and not contained in H; thus, in a sense, we have dualized the notion of the core of H. In this "outer family"<3C(H) of H, we single out certain sub- collections as given in the following definition. 64 65 Definition 3.2: For H a proper subgroup of a finite group G, we define: (1) 066 (H) = {c e OGGDIC 4 G}; (2) one) = {c e Gamma c}; (3) Game“) I {C 6 04G (H)‘C is self-normalizing in G]; (4) 0x16 (11) = {c e o“ (H)‘C is abnormal in G}. Lemma 3.1: Let H be a proper subgroup of a given finite group G, C E OGCH), and D the maximal G-normal prOper subgroup of G. Then: (i) D I corG(C n H). (ii) If G <1 G, then D corG(C n H) C n corGH. (iii) If C I G, then D I corG(C n H) I corGC. m: (i) Since D 4 G and D is properly contained in C, we have, by definition of OGCH), D C H. Thus, D C C n H, and hence D C corG(C n H). 0n the other hand, since corG(C 0 H) 4 G and corG(C n H) is properly contained in C (since C n H I C), we have from the mmcimality of D that corG(C n H) C D. Therefore, the equality in (1) holds. (ii) For C 4 G, we have that C n corGH is normal in G and is con- tained in C n H; thus C n cor H C corG(C n H). But clearly, corG(C n H) C C G since C n H C C; and since corG(C o H) is normal in G and is contained in C n H C H, we have corG(C n H) C corGH so that corG(C n H) C C n corGH. Therefore, corG(C n H) I C n corGH for C 4 G. (iii) Since C n H C C, the inclusion corG(C n H) C corGC is immediate. Now, since C 46 G, corGC is properly contained in C; thus by the maximality of D, corGC C D I corG(C n H). Therefore, corG(C n H) I corGC for C I G. U 66 Using the properties established in the preceding lemma, we can now make the following definitions of the various outer cofactors of a subgrOUp H. Definition 3.3: Let H be a proper subgroup of a finite group G. For C E OG(H)’ we call C/corG(C11 H) an outer cofactor gf'fl in G. More precisely, for C 6 046 (H), we call C/corG(C n H) I C/C n corGH a normal outer cofactor pf.H, If C E Q¥(;(H), we will say that C/corG(C n H) I C/corGC is a nonnormal outer cofactor o_f_ _H_; in particular, if C E O (H), we will say (sn)G that C/corGC is a self—normalizipg outer cofactor pf H, and if C E O>, C/corG(C n S) is Supersolvable, then G is solvable. We will now establish two general theorems from which results parallel to those of the preceding chapter are immediate corollaries. In these two theorems, we shall assume that the trivial group is always a e-group, and hence, in particular, that e-groups do exist. 67 Theorem 3.3: Let 9 be a subgroup-inherited homomorphism-invariant property. If the finite group G has a maximal subgroup S such (1) nonnormal that S and all its (2) self-normalizing outer cofactors are (3) abnormal e-groups, then cofGH I H/corGH is a e-group for all prOper (l) nonnormal (2) self-normalizing subgroups H of G. (3) abnormal IPpggf: Let H be any proper (k)-subgroup of G, where k I l, 2, or 3 (that is, (k) denotes one of the three prOperties (1) nonnormal, (2) self-normalizing, or (3) abnormal). Now, if H C S, then trivially H/corGH is a e-group since S is a e-group and e is subgroup-inherited and homomorphism-invariant. Thus, suppose H C S. If corGH C S, then H is an element of GIG (S), 0 (S), or G>4G (S) according as k I 1, (sn)G 2, or 3 so that, by hypothesis, H/corGH is a e-group. If corGH C S, then S corGH I G by the maximality of S. In this case, since S is a G-group and 9 is homomorphism-invariant, G/corGH I S corGH/ corGH which is isomorphic to 8/8 0 cor H also is a e-group; hence, since 9 is sub- G group-inherited, H/corGH is a e-group. U Note: The proof of the preceding theorem also shows that if S and all its (k)-outer cofactors C/cor C, with C a maximal subgroup of G G, are e-groups, then H/corGH is a e-group for all (k)-maximal sub- groups of G. The following results are now immediate consequences of the pre- ceding theorem or the note, Lemma 1.2, and the corresponding results of Chapter 2. Part (b)-(vi) strengthens Theorem 3.2 by removing the condition "or C n S I-< l>" and by giving information about the Fitting lengths of G' and C. These results also show that there is nothing Corollary 3.4: conditions. 68 eSpecially significant about the condition of Supersolvability imposed in Theorem 3.2, but that it can be replaced by a variety of other Suppose that the finite group G has a maximal subgroup S for which one of the following nine conditions holds: (a) S and all its abnormal outer cofactors T/corGT with T a maximal subgroup of G are (i) (ii) (iii) (b) S and (iV) CV) (Vi) (c) S and (vii) (viii) (1X) nilpotent. nilpotent of class I n. solvable of derived length E n. all its self-normalizing outer cofactors are p-nilpotent. o-Sylow-towered for a some fixed ordering of a set 2 of primes containing n(G). supersolvable. all its abnormal outer cofactors are p-nilpotent, and either p is odd or the p-Sylow subgroups of G are abelian. C-Sylow-towered for a some fixed ordering of a set 2 of primes containing n(G), and the 2-Sylow subgroups of G are abelian. supersolvable, and either the 2-Sylow subgroups of G are abelian or the abnormal maximal subgroups of G all have prime-power index. Then, in the reapective cases, the following hold: (a) (i) G is solvable with G/F(G) nilpotent. 69 (ii) G is solvable with yn(G) nilpotent (and the other conclusions of Theorem 2.7 hold). (iii) If G is solvable, then G is nilpotent (and the other conclusions of Theorem 2.9 hold). (b) (iv) G has a normal p-subgroup P0 (which may be trivial) such that G/P0 is p-nilpotent; in particular, G is p-solvable of p-length § 2. (v) G is solvable with G/F(G) o-Sylow—towered. (vi) G is solvable with G/F(G) supersolvable and G/F2(G) abelian; thus §(G') § 2, §(G) E 3. (c) (vii) Same as (iv). (viii) Same as (v). (ix) Same as (vi). Theorem 3.5: Let 0 be a subgroup-inherited homomorphism-invariant property. Then the following are equivalent: (a) For all abnormal maximal subgroups S of G, the (2) abnormal {31) self-normalizing:} outer cofactors of S are e-groups. (b) H/corGH is a B-group for all {(1) self-normalizing} (2) abnormal subgroups H of G. Ppggf: (b) a (a) is immediate. (a) a (b): Suppose this to be false, and let G be a minimal counterexample. Then there exists some proper (k)-subgroup H I <11) of G such that H/corGH is not a G-group, where k I l or 2. Suppose first that corGH =.< l>. Then H d F(G) I the intersection of all abnormal maximal subgroups of G. For, by Lemma 2.41, F(G) is a normal nilpotent subgroup of G, and if H C F(G), then Hi4 4 F(G) 4 G, hence H 4'4 G; however, if H is a (k)-group, it cannot be subnormal ‘!lll|lllllll 70 in G. Thus, H C F(G), and there exists an abnormal maximal Subgroup S (S) of G not containing H. Since corGH I < l> C S, H belongs to o(sn)G or OXG (S) according as k I l or 2, so that H/corGH is a e-group by hypothesis. This, however, contradicts the choice of H. Suppose now that corGH I < l> and consider G I G/corGH. We assert that (a) holds for G. For this, let S I S/corGH be any abnormal maximal subgroup of G, and let G I C/corGH be any element of O (sn)"(s) or ONE (S) according as k I 1 or 2. Then S is an abnormal maximal subgroup of G, and C is a (k)-subgroup of G; also, C Si S since C C S. Now, from Ienuna 1.4, we have corE(C) I corGC/corGH. Since coraaf) C S- (from the definition of an outer cofactor of S), it follows that corGC C S, and thus that C belongs to (S) or 0 NC (S) according C’(sn)G _as k I l or 2. By hypothesis, therefore, C/corGC is a e-group, and hence so also is C/corE(E) I C/corGC (by lama 1.4). Since (a) thus holds for G and ‘5‘ < ‘G‘ , it follows from the minimality of G that (b) also must hold for G, that is, the cofactors of. all proper (k)-subgroups of G are e-groups. In particular, H I H/corGH is a proper (k)-subgroup of G and core-(H) I so that H I H/coradl) is a e-group. However, this again contradicts the choice of H. We conclude, therefore, that no such minimal counterexample can exist; and the result now follows. ‘1 11959: An obvious modification of the preceding proof shows that the following are equivalent: (a) For all abnormal maximal subgroups S of G, the (k) -outer cofactors T/corGT of S with T a maximal subgroup of G are e-groups. ‘Illillllll‘l‘ 71 (b) H/corGH is a e-group for all (k)-maximal subgroups H of G. CorreSponding to Corollary 3.4, we have the following results which are direct consequences of the preceding theorem or the note, Lemma 1.2, and the corresponding results of Chapter 2. Corollary 3.6: Suppose that for each abnormal maximal subgroup S of the finite group G, one of the following nine conditions holds: (a) The abnormal outer cofactors T/corGT of S with T a maximal subgroup of G are: (i) nilpotent. (ii) nilpotent of class E n. (iii) solvable of derived length I n. (b) The self-normalizing outer cofactors of S are: (iv) p-nilpotent. (v) o-Sylow-towered for a some fixed ordering of a set 2 of primes containing n(G). (vi) supersolvable. (c) The abnormal outer cofactors of S are: (vii) p-nilpotent, and either p.is odd or the p-Sylow sub- groups of G are abelian. (viii) o-Sylow-towered for a some fixed ordering of a set 2 of primes containing n(G), and the 2-Sylow subgroups of G are abelian. (ix) supersolvable, and either the 2-Sylow subgroups of G are abelian or the abnormal maximal subgroups of G all have prime-power index. 1 Then, in the reapective cases, the conclusions (a)-(i) through (c)-(ix) of Corollary 3.4 hold. ‘l-Illllul'llllllII-llllllll'l 72 3.3 Influence on a Group of the Normal Outer Cofactors of Subgroups We turn now to a consideration of the normal outer cofactors of the maximal subgroups of a finite group G and investigate what effect properties imposed on these will have on G. Such an approach is sug- gested by Deskins in [5]. As mentioned there, while every maximal sub- group of a finite solvable group has prime-power index, the converse is not true, as the simple group of order 168 shows. Deskins then defines the normal index of a maximal subgroup; it is precisely this that must be of prime-power for the group to be solvable. Let us recall that for a maximal subgroup S of a finite group G, 04 G(S) consists of all those subgroups H CG which satisfy (1)11 d s, that is, HS = G, (2) Hi4 G (where we allow H I G), and (3) L C S, that is, LS I S, for all prOper G-normal subgroups L of H. Also, the normal outer cofactors of S are the groups H/corG(H.n S) I H/Ht) corGS with H,€ G;(;(S). Using this terminology and notation, we can state the theorem of Deskins, which makes possible the defin- ition of normal index, as follows. Theorem 3.7: Let S be a maximal subgroup of the finite group G. Then: (i) All normal outer cofactors of S have the same order. (ii) If ‘G:S‘ I a power of a prime, then there exists a unique H 6 04G (S). Definition 3.4: The normal £292! of a maximal subgroup S of a finite group G iS the order of any normal outer cofactor of S. The following theorem extends statement (i) of the preceding theorem and also shows that if we impose a condition on one of the ['I“'.llhlil..ll.ll..‘lllllliIlllll'lls‘]i[.l[ i .I‘.. . 73 normal outer cofactors of a maximal subgroup S, then, in fact, we are imposing it on all of them. Theorem 3.8: Let S be a maximal subgroup of a finite group G. Then all the normal outer cofactors of S are isomorphic. M: Let H and K be distinct elements of 04G (8). We are to show that H/H n corGS and K/K n corGS are isomorphic. Case 1: corGS I < l>.—In this case, H and K are minimal normal subgroups of G. For suppose L 4 G with L Ci? H. Then, Since H 6 04G (S), we have L C S; and since L 4 G, this means that L C corGS I < l>, so that L I <1 >. Consequently, H contains properly no nontrivial normal subgroup of G, and is therefore minimal normal in G. Similarly, K is shown'to be minimal normal in C. It follows then that H and K centralize each other. Also, we have H n S I K n S I < l>. For Since H is normal in G, H n S is normal in S, and thus S C NGCH f) 8). Now, K centralizes H, hence centralizes H n S so that K C NG(H n S) also. Consequently, G I KS C NCO-i n S), that is H n S 4 G. But since corGS I, this means that H n S I < l>. In the same way we obtain K n S I < l>. Now, from Dedekind's Law, HCHK n S) I HK 0 HS I HK n G I HK, and K(HK O S) I HK 0 KS I HK n G I HK also. It follows that H =H/Hn KIHK/KIKO-IKn S)/KI (mm S)/(m.—In this case we consider G I G/corGS, GS, H I H corGS/corGS, and K I KcorGS/corGS. We show first that H and K belong to 045- (S). and we let S I S/cor For this, suppose that L I L/corGS is a normal subgroup of G which is preperly contained in H. Then, since H n L 4 G, H n L CF- H, and 74 H (_- 04G (S), we have that H O L C S, and hence H n L C corGS. But since corGS C L, we have H ’1 corGS C H n L; thus, H n L I H n corGS and ‘H“cor S‘ ‘HL‘L‘ g ‘H 01‘ I ‘H n corGS‘ = ‘H corGg‘ And since HL I H(L corGS) I (H corGS)L I Hcor GS, it follows that ‘L‘ I ‘corGS‘ so that L I . This shows that H E 0456); and in a similar manner, one shows that K 6 (£156). Now, Since coraé) I , it follows from Case 1 that H I K. Therefore, H/H 0 cores I H'I K‘I K/K.n corGS. U Statement (iii) of the following result coincides with statement (ii) of Theorem 3.7. We include a proof of it here since none is given in [5]. Theorem 3.9: Let S be a maximal subgroup of a finite group G. Then: (i) The normal outer cofactors of S are p-solvable if and only if the normal index of S is either a power of p or is prime to p. (ii) The normal outer cofactors of S are solvable if and only if the normal index of S is a power of a prime. Then, the normal index of S I the index of S. (iii) If S has prime-power index, or prime-power normal index, then there exists a unique H in O4G(S) - 'ggggg: (i) Suppose first that the normal outer cofactors of S are p-solvable, and let H E(} (S). As in the proof of Theorem 3.8, 4G 8 6 046(3) where S— I S/cor S and G I Clear 8; and G G since corE-(S) I <-l->, H is minimal normal in 6. Now, by hypothesis, H I H corGS/ corG H/H n corGS is p-solvable, hence so also is H I H/H n corGS. Thus, H is either a p-group or a p'-group so that the normal index of S I ‘H/H n corGS‘ I ‘H‘ is either a power of p or is prime to p. 75 The converse implication is an immediate consequence of the definition of normal index. (ii) The equivalence of the two statements given here follows from (i) together with the fact that a finite group G is solvable if-f G is p-solvable for all primes p dividing ‘G‘. For the second part, let H 6 04G(S) with H/H n corGS solvable. For E, S, and H as above, we have H r—j 0456) and H is minimal normal in G. Since H'I H/H n cor S, H'is solvable, hence is elementary abelian. Now G n S 4 S since H4 G; and H n S 4 H since H is abelian; thus, we have ml H 2‘) S4 HS I G. But corE(S) I <1> so that H n S I . Therefore, ‘G:S‘ I ‘G:S‘ I ‘HS:§‘ I ‘H:Hn S‘ I ‘H‘ I ‘H/H n corGS‘ I the normal index of 8. (iii) We note first that if H G O4G(S)’ then H/H n corGS is divisible by ‘H:H n S‘ I ‘HS:S‘ I ‘G:S‘, so that the index of S divides the normal index of S. If the normal index of S is a power of a prime, therefore, so also is the index of S. Thus, suppose that S has prime- power index, say ‘G:S‘ I pa; and let H (1 O4G(S) . Case 1: corGS I < l>.—In this case, H is minimal normal in G, as shown in the proof of Theorem 3.8. If CG(H) I < l>, then H is the unique element of 04 (S). For if K 6 046(8) and H I K, then K also G is minimal normal in G so that H 0 K I < l>, and H and K centralize each other; thus K C Cam) I < l>, which is impossible since K E O4G(S) implies that K C S. 80 suppose that C I CG(H) I < l>. Since C 4 G, we have C C S since cares I < 1), so that OS I G. Then H n S I < 1). For Since H 4 G, we have H n S 4 S; also, C C NG(H n 8) since C centralizes H, 76 hence centralizes H n 8. Thus G I CS : NGOI n S), that is H n 8 <1 G. But since corGS I < l>, this means that H n S I < l>. Therefore, ‘H‘ I ‘H:H n S‘ I ‘HS:S‘ I .‘G:Sl I pa so that H is solvable, hence elementary abelian, and H c C I CC (H). It follows from this that C n S I < l>. For C n S is normal in S, and H centralizes C, hence centralizes C n S; thus G I HS C NG(C n S), that is, C n S 4 G. However, cor S I <1>, so that C n S I. G It now follows by Dedekind's Law that H=H(CDS)=CHHS=CnGIC. This then implies that H is the unique element of 04 (S). For any G other K E O4G(S) would centralize H as above, which would contradict H I C601). Case 2: corGS 9‘ < l>.—In this case, we let G I G/corGS and S I S/corGS. Now let H F O4G(S)' As before, H I HcorGS/corGS E 045(S)' Thus, by Case 1, since ‘G:S| I ‘G:S‘ I pa and corE(S-) I , H is the unique element of 0456)’ Now suppose there exists K E 046(8) with K 3‘ H. Then, as before, K I KcorGS/ cor G contradicts the uniqueness of H, and thus establishes the result. For S G 0456). We will show that K 3‘ H, which this, suppose that K I H so that H corGS I KcorGS. Let x I hk, where l h E H, 1; € K, be an arbitrary element of HR. Then h- s- I ks for 1 2 . _ _ -1 . 2 belonging to corGS. Hence, x hk sls2 C corGS. This shows then that HK C corGS. But this is impossible, since H (1 Gd implies that H i 3. Therefore, K 3‘ H as we wished to show. I] some s1, 3 S G( ) Deskins has shown in [S] the following equivalences. Theorem 3.10: For G a given finite group, the following are equivalent: (1) G is solvable. 77 (ii) Each maximal subgroup of G has prime-power normal index. (iii) The index and normal index are equal for each maximal subgroup of G. Our next two theorems are extensions of this result. The following lemma will prove useful in establishing not only these two theorems but later results as well. Lemma 3.11: Let M be a normal subgroup of the given group G and 8/14 a maximal subgroup of G/M. Then each normal outer cofactor of S/M is isomorphic to every normal outer cofactor of S. m: Let E I G/M, S I S/M, and let El? 0 core-(S) be a normal outer cofactor of S, where K I K/M. We have iumediately that S is a maximal subgroup of G, K 4 G, and K ¢ S. It follows that L C: K for some L E O4G(S)' Now, ML = K. For ML c. K so that if = ML/M c K/M = E; also, K4 5, and HI ¢ S- (since L 93 8). Thus, since K € 045(3) , we have 1.4: I K, and hence, ML I K. Since M C cor S, it follows that L/L n cor S 7‘ LcorGS/ corGS I lMcorGS/ corGS G I K corGS/ corGS °-‘ K/K n corGS; and from Lemmas 3.1 and 1.4, we have K/M K/K fl cor S I K/corG(K n S) - (:0er nS)/M G I K/corC-(‘K n S) I K/K q corC.(S). Therefore, the normal outer cofactor K/K n corc(S) of S I S/M is isomorphic to the normal outer cofactor L/L n cor S of S, and hence, G by Theorem 3.8, to every normal outer cofactor of S. U 78 Theorem 3.12: For a finite group G, the following are equivalent: (i) G is p-solvable. (ii) G has a maximal subgroup S such that S and its normal outer cofactors are p-solvable. (iii) For each abnormal maximal subgroup S of G, the normal outer cofactors of S are p-solvable. (iv) For each abnormal maximal subgroup S of G, the normal index of S is either a power of p or is prime to p. N232; As the proof shows, the word "abnormal" can be omitted in (iii) and (iv). 2522; gfi Theorem §;lgg (i) a (ii) is trivially true. (ii) a (i): We use induction on ‘G‘. Let S be a maximal sub- group of G which is p-solvable, and let K 6 046(8) with K/K n corGS p-solvable. Case 1: K.n corcs I < 1>u-—Then K is a minimal normal subgroup of G. For if L is a normal subgroup of G with L, hence L I < l>. Now, by hypothesis, K I K/Kffi cor S is p-solvable. This implies G that since K is minimal normal in G, K is either a p-group or a p'-group. Since KS I G, we have G/K I KS/K I 8/8 D K is p-solvable, since S is p-solvable. Therefore, since K is either a p-group or a p'-group, G is p-solvable. Case 2: K n cor S 3‘ < l>.—Let M be a minimal normal subgroup G of G contained in K.n corGS. Since MI: S, M is p-solvable, and hence is either a p-group or a p'-group. 79 Now, (ii) holds for G/M. For S/M is a p-solvable maximal sub- group of G/M; and by Lemma 3.11, each normal outer cofactor of S/M is isomorphic to the normal outer cofactors of S, hence is p-solvable. By induction, therefore, G/M is p-solvable. And since M is either a p-group or a p'-group, it follows that G is p-solvable. (i) -+ (iii) is clear. (iii) a (i): If G is simple, then every maximal subgroup of G is abnormal in G. Let S be any such, and let H G 04 (S). Then G H ¢ 8, H 4 G, and G simple imply that H I C so that 046(8) I {G}. Therefore, since corGS I‘, G I G/G n corGS is p-solvable. So suppose that G is not simple and let M be a minimal normal subgroup of G. Then M,is p-solvable, hence is either a p-group or a p'-group. For if Ml: F(G) I the intersection of all abnormal maximal subgroups of G, then since F(G) is nilpotent by Lemma 2.41, M is nil- potent, thus is p-solvable. On the other hand, if M ¢ F(G), then there exists an abnormal maximal subgroup S of G which does not contain M. Then M 6 046(8) 80 that M I M/M fl corGS is p-solvable by hypothesis. Now consider 6 I G/M. Then (iii) holds for G: For let E": S/M be any abnormal maximal subgroup of G, and let K- I K/M 6 0d a(g); we must show that K/K n cor-5(5) is p-solvable. S is clearly an abnormal maximal subgroup of G; by hypothesis, therefore, all normal outer co- factors of S are p-solvable. By Lemma 3.11, K/K n corE(S) is isomorphic to the normal outer cofactors of S, and hence is p-solvable. The hypotheses thus hold for E = G/M so that, by induction, G/M is p-solvable; and since M is either a p-group or a p'-group, it follows that G is p-solvable. (iv) ... (i) is immediate since (iv) clearly implies (iii). /— 80 (i) ... (iv): Let G be p-solvable, S a maximal subgroup of G, and K 6 04G(S). To show that K/K n corGS is either a p-group or a p'-group, it suffices to show that it is a chief factor of G. And this is immediate. For if L <1 G with K n cor s c L Ci K, then L c s G since K E O4G(S)’ and thus L c corGS since L 4 G. Therefore L : Kn corGS so that L I K n corGS.fl Theorem 3.13: For a finite group G, the following are equivalent: (i) G is solvable. (ii) G has a maximal subgroup S such that S and its normal outer cofactors are solvable. (iii) For each abnormal maximal subgroup S of G, the normal outer cofactors of S are solvable. (iv) For each abnormal maximal subgroup S of G, the normal index of S is a power of a prime. (v) For each abnormal maximal subgroup S of G, the normal index of S I the index of S. (vi) For each abnormal maximal subgroup S of G, K n S is normal in G for all K G O4G(S)° (vii) For each abnormal maximal subgroup S of G, K n S is subnormal in G for all K E O4G(S)‘ Eggs: As in the preceding theorem, the proof shows that the word "abnormal" can be omitted in (iii)-(vii). £5922 9_f Theorem §__]_._§_: The equivalence of- (i)-(iv) is imediate from Theorem 3.12 and the fact that a finite group G is solvable if-f G is p-solvable for all primes p dividing ‘G‘. (i) .. (v) follows from Theorem 3.9. 81 (v) _. (i): If G is simple, then all the maximal subgroups of G are abnormal in G. Let S be any such and K C O4G(S)' Then K 4 G, K i S, and G simple imply that K I G, and thus that O4G(s) I {G}. Also, corGS I < l> since G is simple. By hypothesis, \G:S‘ I the normal index of S I ‘G/G n corGS| I ‘6‘ so that ‘8‘ I 1. Therefore, G has no nontrivial maximal subgroups, hence is cyclic of prime order, and thus is solvable. So suppose G is not simple, and let M be a minimal normal sub- group of G. Then (v) holds for G/M. For if S/M is an abnormal maximal subgroup of G/M, then S is an abnormal maximal subgroup of G; and by Lemma 3.11, each normal outer cofactor of S/M is isomorphic to the normal outer cofactors of 8 so that the normal indices of SIM and S are equal. Thus, lG/M:S/M‘ I ‘G:M‘ I the normal index of S =‘ the normal index of S/M. By induction, therefore, G/M is solvable. If G has a minimal normal subgroup M* i5 M, then G/M* is likewise solvable by induction, hence so also is G/M X G/M*. Since G I G/M n M* is isomorphically embedded in G/M x G/M*, it follows that G also is solvable. We may assume, therefore, that M is the unique minimal normal subgroup of G; and we need only show that M is solvable. This is clear if M c {>(G); so suppose that M ¢ §(G), and hence that there exists a maximal subgroup of G not containing M. Let the prime p divide ‘M‘ . Now, if S is maximal in G and M ¢ S, then S J G by the uniqueness of M so that S is abnormal in G; by hypothesis, therefore, since M'E 046(8)’ ‘G:S| I the normal index of S I ‘M/M n corGS‘ I ‘M‘, so that pHG:S|. Thus M c n {S‘s is a maximal subgroup of G with p“G:S‘}; and this latter group is a normal solvable subgroup of G 82 (see, for example, Deskins [51). Consequently, M is solvable, and the result follows. (v) 0 (vi): Let S be a maximal subgroup of G and K E G4G(S)° Then, KflS4G if-f Kn corGS IcorG(Kn S) IKflS if-f \K/K n corcsl = |K:K n s| (since K n corGS s: K n S) if-f 1K/Kn corGS‘ |Ks:s| = |c:s\ if-f normal index of S I index of 8. (vi) .. (vii): We show first that if T is any proper subgroup of G and K E O4 6 and s empm}. 11953: In view of Theorem 3.9-(i), the collection mp(G) could as well be defined as the family of maximal subgroups whose normal outer cofactors are not p-solvable. mg; Theorem 3.15: We show (i) Rp(G) = n {3‘s 6 mp(c)}, and then (ii) Rp(G) = n {S‘s >< G and s emp(c)}. (i) Let Tp(G) I f) {S|S E 771p(G)}; we wish to show that Rp(G) I Tp(G). We note first that every p-solvable minimal normal subgroup M of G (which is thus either a p-group or a p'-group) is contained in Tp(G). For suppose M i Tp(G) for such an M; then there exists S E ”(F(G) with M ¢ 8, and thus, M i cor S. Now, M E O4G(S) G so that the normal index of S I ‘M/M n corGS‘ I |M| , and hence ‘M‘ is divisible by both p and some prime I p. Thus, M is not p-solvable. It follows then that if TP(G) I < l>, then RP(G) I < 1) also, so that Tp(G) I Rp(G). We may assume, therefore, that Tp(G) I < l>. Now, TP(G) is a normal subgroup of G. For, from the definition of O4G(S) for S a maximal subgroup of G, if x E G, then H E Gd (S) G if-f H 6 O4G(Sx); thus, the normal index of S I the normal index of 8". Therefore, S 6 7RP(G) if-f Sx 6 mp(G) for each x G C. Now, let M be a minimal normal subgroup of G contained in TP(G). We show first that M is p-solvable, hence is either a p-group or a p'-group and is contained in Rp(G). For suppose that M is not 84 p-solvable. Then M is not contained in the nilpotent subgroup F(G) I the intersection of all abnormal maximal subgroups of G. Thus there exists an abnormal maximal subgroup S of G not containing M. As above, M e O4G(S) and the normal index of S I ‘M‘ . Since M is not p-solvable and hence is neither a p-group nor a p'-group, there exists a prime q I 1) such that both p and q divide ‘M‘ . Then both p and q divide the normal index of S so that. S E 'mp(G). And since M d S, this means that M ¢ Tp(G) . Now, Rp(G/M) I Rp(G)/M. For since Rp(G) is a normal p-solvable subgroup of G containing M, we have Rp(G)/M is p-solvable and normal in G/M; thus Rp(G)/M c Rp(G/M). 0n the other hand, if Rp(G/M) = K/M, then K is normal in G, and K is p-solvable since K/M is p-solvable and M is either a p-group or a p'-group. Thus K c RP(G) so that Rp(G/M) I K/M : Rp(G)/M, which establishes the equality. Also, we have Tp(G/M) I Tp(G)/M. For as was shown in the proof of Theorem 3.11, if S/M is a maximal subgroup of G/M, then the normal outer cofactors of S/M are isomorphic to the normal outer cofactors of S and the normal index of S/M I the normal index of S. And since M s: TP(G), we have M s: S for all S E mp(G). It follows, therefore, that s e mpm.) if-f S/M e ‘mp(G/M). Thus, mp(c/M) :- {s/u‘s e mp(c)}. so that Tp(G/M) = n {s/u‘s 6 mm} = [n {s|s e mp}]/M = Tp(G)/M. Now, by induction, Tp(G/M) I Rp(G/M). By what we have just shown, therefore, Tp(G) /M I Rp(G)/ M, and hence, TP(G) I Rp(G). (ii) Let pr) I 0 {8‘8 >4 G and S 6 7RD(G)}; we wish to show now that TP(G) I Rp(G). First, it is immediate from the definitions 85 that Tp(G) C TP(G); thus, if $P(G) I < l>, then Tp(G) I < l>, and from (i), RP(G) I < 1> so that a(a) = Rp(G)' So _suppose that Tpm) I < l>. Tpm) is a normal subgroup of G. For from (i), we have S G 'IRP(G) if-f Sx é 'Inp(G) for each x E G; and from the definition of abnormality, it is clear that 8 >4 G if-f 5" >4 c for each x e G. Now let M be a minimal normal subgroup of G contained in Tpm). From (i), Rp(G/M) I Rp(G)/M. Also, we have Tp(G/M) I Tp(G)/M. For letting fipm) I {813 ><1 G and S E mp(G)}, we have from the proof of (i) that 7’7'zP(G/M) = {s/M\s/M >4 G/M and s e mpm}; and since it is immediate from the definition of abnormality that s/M >4 G/M if-f 3 >4 G and M c s, we have filp(G/M) = {s/M|s >4 c, s e mp(c), and M c s} = {S/M‘S e fipm) and M c 8}. But since M c TP(G), we have that M c S for all S 6 51pm); thus MPG/M) I {S/M|S 6 51pm”. Hence, Tp(G/M) = n {s/M|s e ’ilp(c)} = [n {s|s e 77(p(G)}]/M = Tp(G)/M. Now, by induction, Rp(G/M) I l‘].?'p(G/M); therefore, Rp(G)/M I Tp(c)/M, that is, RP(G) = Tpm). [1 Corollary 3.16: Let R(G) I the largest normal solvable subgroup of G and m(G) I the collection of all maximal subgroups of G with normal index divisible by two distinct primes. Then, R(G) -= n {s|s 6 mm} = n {s|s >4 c and s e 77((G)}- / H _L I . 86 Note: As in the theorem, we can, by Theorem 3.9-(ii), let 772(G) be the family of all maximal subgroups whose normal outer cofactors are nonsolvable. Proof of Corollary: Let n(G) I {p1, p2,... , pm}. From the fact that a group H is solvable if-f it is p-solvable for all primes p m dividing ‘Hl, it follows that R(G) I n Rp (G). Now, for each i=1 i i I l, 2, ..., m, let mp (G) I {Sijlj I l, , ni}. Then, from the i preceding theorem, we obtain an m npi R(G) = n R (G) = n n s.. =n {S\S 6772(6)}- i=1 pi i=1 j=1 13 In a similar manner, R(G) I n {sls >6 G and S 6 771(6)}. [1 3.4 Addendum Since the preparation of this thesis, two papers relevant to the results given here have come to my attention. The first of these, "The g-normalizers of a finite soluble group" by R. Carter and T. Hawkes, appears in the Journal of Algebra, 5 (1967), 175-202, and is a study of finite solvable groups from the viewpoint of the theory of formations. Although there is little actual overlap with this thesis, some of the results are related to those of this thesis. For example, the equivalence of the first two statements of Theorem 2.9 is a con- sequence of one of the results of this paper. And since the normal outer cofactors of a subgroup of a finite group G are chief factors of G, the theorems of Carter and Hawkes dealing with chief factors of a finite solvable group have some bearing on the results of Chapter 3 and, more particularly, on those of Section 3.3. The second paper is that of J. Beidleman and A. Spencer, "The normal index of maximal subgroups in finite groups", which, as of this 87 date, has not yet been published but has been submitted to the Illinois Journal of Mathematics. The results are closely related to those of Section 3.3. There are, for example, statements which can be added to the lists of equivalent statements in Theorems 3.12 and 3.13. BIBLIOGRAPHY l. 2. 10. ll. 12. 13. 14. 15. BIB LIOGRAHlY R. Baer, "Classes of finite groups and their properties," Illinois Jour. Math., 1 (1957), 115-187. W. 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