FITTING SETS 1N FINITE SOLVABLE GROUPS Thesis for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY L. B. WADE ANDERSON, JR. 1973 k .— a» . ‘ LIBK XE 1‘; k" ,L Michigan State > Unzvcrsrty f,“ w This is to certify that the thesis entitled FITTING SETS IN fi- FINITE SOLVABLE GROUPS 1 presented by L. B. WADE ANDERSON, JR. has been accepted towards fulfillment of the requirements for Ph.D. degrwin Mathematics Major 0 ssor Date JUIY 25! 1973 0-7639 ABSTRACT FITTING SETS IN FINITE SOLVABLE GROUPS By L.B. Wade Anderson, Jr. Let G be a finite solvable group. A Fitting set in G is a collection 3 of subgroups of G satisfying: i) S E 3 and N a normal subgroup of 8 implies N E 3, ii) H,K G 3 and H,K normal subgroups of HK implies HR 6 3, iii) S 4 ? implies Sx E 3 for all X E G. This is a localization of the Fitting classes introduced by Fischer, Gaschutz and Hartley in 1967 [Math Z., 102 (1967), 337-339]. In particular, a Fitting class gives rise to a Fitting set in each solvable group G. Several results which appear in the literature as pertaining to Fitting classes are obtained as consequences of the local theory. In addition, local considerations expose some previously unknown facts. If 3 is a collection of subgroups of G, a subgroup V of G is called an 3-injector of G if V fl'N is 3-maximal in N for each subnormal subgroup N of G. Theorem (Fischer, Gaschutz, Hartley): If 3 is a Fitting set in G, then G possesses 3-injectors and all 3-injectors of G are conjugate in G. Theorem: Let 3 be a Fitting set in G and V an 3-injector of G. A subgroup w of G is an 7-injector of G e W E 3 and V and W cover and aovid precisely the Same chief factors of G. L.B. Wade Anderson, Jr. A subgroup V of G is p-normally embedded in G if there is a normal subgruip N of G and a sylow p-subgroup P of G such that P H N is a sylow p-subgroup of V. If V is p—normally embedded in G for all primes p, V is normally embedded in G. Theorem: Let 3 be a Fitting set in G and V an 3-injector of G. If there exists a normal Subgroup N of G contained in V such that V/N is nilpotent, then V is normally embedded in G. Theorem: A subgroup V of G is normally embedded in G a [SX\S s V, x E G} is a Fitting set in G. FITTING SETS IN FINITE SOLVABLE GROUPS By L.B. Wade Anderson, Jr. A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1973 . d“ «Whig (fik ACKNOWLEDGEMENTS The author wishes to express his appreciation to Professor Ti Yen for his many helpful suggestions and patient guidance during the research. He would also like to thank his wife, Pat, and children Wade, Stan and Lynn for their tolerance and encouragement. ii TABLE OF CONTENTS Page INTRODUCTION 1 Chapter I FITTING SETS 4 §l. Definitions and Basic Properties; g-radicals 4 §2. 3-injectors 8 §3. Fitting Classes 14 §4. Fitting Sets and Direct Products 17 §5. Fitting Sets and Quotients 18 II FITTING SETS WITH NORMALLY EMBEDDED INJECTORS 23 §l. Normally Embedded Subgroups 23 §2. Fischer Sets 26 §3. More On Normally Embedded Subgroups 32 §4. Normal Fitting Sets 38 III INJECTORS 41 §l. Cover—Avoidance Characterizations of Injectors 41 §2. Injector Subgroups 43 BIBLIOGRAPHY 49 NOTATION AND SYMBOLS Groups are denoted by large Latin letters, group elements by small Latin letters and collections of groups by German script. 6, ; have their usual set theoretic meaning. Symbols used most often are: \G\ the order of G H s G H is a subgroup of G H < C H S G and H # G N S G N is a normal subgroup of G G/N the factor group of G by N NG(H) the normalizer of H in G CG(H) the centralizer of H in G the subgroup generated by V and W HG coreG(H) fl Hx xEG Sylp(G) the set of sylow p-subgroups of G n' the set of all primes p é n, where n is a set of primes 6 the class of all n-groups, where n is a set of primes W the class of nilpotent groups On(G) the largest normal n-subgroup of G N 9 S G N is subnormal in G iv INTRODUCTION In all that follows g£922_will always be understood to mean finite solvable group. A Fitting set in a group G is a collection 3 of sub- groups of G satisfying: i) S E 3 and N a normal subgroup of S implies N E 3, ii) H,K G 3 and H,K normal subgroups of HK implies HK E 3, iii) S E 3 implies 8X 6 3 for all x E G. This is a localization of the Fitting classes introduced by Fisher, Gaschutz and Hartley in [3]. In that paper a Fitting class is defined as an isomorphism closed collection 3 of groups satisfying i) and ii) given above. In particular, every Fitting class 3 induces a Fitting set in each group G which consists of those subgroups of G belonging to 3. The goal here is to develop and investigate some of the consequences of this local theory. Many results pertaining to Fitting classes which appear in the literature are obtained here as consequences of the local theory. In all cases, these known results are more readily deduced using Fitting sets. In addition, local considerations have exposed several previously unknown facts which pertain to Fitting classes, and these facts appear extremely difficult to establish without use of the local theory. If 3 is a collection of subgroups of G, a subgroup V of G is called an 3-injector Of G provided that V FlN is an 3-maxima1 subgroup of N for each subnormal SubgrOUp N of G. The major result of [3] is that for a Fitting class 3, every group G possesses 3-injectors and all 3-injectors of G are conjugate in G. Close examination reveals that this is actually a local theorem. That is, the result holds for any Fitting set in G. A proof of this is included in Chapter I, but differs only slightly from that given in [3]. For the most part, Chapter I is devoted to the development of the basic theory of Fitting sets. A crucial section is §5 of Chapter I. There the quotient aspects of the theory are developed and these constitute the basic tools utilized throughout the remainder of the work. A subgroup V of G is said to be p-normally embedded in G if a sylow p-subgroup of V is a sylow p-subgroup of some normal subgroup of G. If V is p-normally embedded in G for all primes p, then V is normally embedded in G. Chapter II deals with Fitting sets 3 in G having the property that the 3—injectors are normally embedded Subgroups. In a first attempt to dualize W. Gaschutz's Formation Theory, B. Fischer introduced what are now called Fischer 3-subgroups and Fischer classes. For a collection of groups 3 a Fischer 3-subgroup Of G is a subgroup V of G such that V E 3 and V contains every 3—subgroup of G which it normalizes. A Fischer class is a Fitting class 3 that also satisfies: (*) N S S E 3, N S T s S and T/N has prbne power order implies T E 3. In [4], B. Hartley establishes the following facts: 1) If 3 is a Fischer class, the 3-injectors of G are normally embedded subgroups of G. 2) If 3 is a Fischer class, the Fischer 3-subgr0ups of G are precisely the 3—injectors of G. In §2 of Chapter II it is seen that these facts are also consequences of the local theory. Using the local version Of 1) it is shown that if 3 is any Fitting set in G and the 3— injectors of G are nilpotent, then theyare necessarily normally embedded Subgroups of G. Fact 2) above appears here as a corollary to the local theorem which states that if the 3-injectors are normally embedded then a subgroup W of G is an 3-injector if and only if W contains each 3-subgroup of G which it normalizes. In §3 there are given necessary and sufficient conditions in order that a subgroup be normally embedded, the major result being that the normally embedded subgroups of G are characterized as those subgroups V of G for which {SX\S S V, x E G} is a Fitting set in G. Chapter II, §4 contains a local proof of a theorem due to Blessenohl and Gaschutz concerning normal Fitting classes ([1], Satz 6.2). In Chapter III it is shown that among the members of a Fitting set 3 in G, the 3-injectors of G are characterized by their cover-avoidence property. There is also included a dis— cussion of the collection Of all those subgroups of G which are injectors of G for some Fitting set in G. CHAPTER I FITTING SETS In this chapter the theory of Fitting sets is introduced and developed. The theory of Fitting sets is a localization of the theory of Fitting classes introduced by Fischer, Gaschutz and Hartley in [3]. Several statements appearing in §1 and §2 are localizations of results which can be found in [3] and [4] as pertaining to the theory of Fitting classes. An investigation of the proofs of these reSults shows that they are actually local in nature. Proofs of such statements are included here for the purpose of completeness but differ at most only slightly from those given in [3] and [4]. In §3 Fitting classes are discussed in terms of Fitting sets and the above mentioned global results are included as corollaries of the corresponding local theorems. The quotient aspects of the theory of Fitting sets are con- sidered in §5 and it is here that the major tools are developed which are basic for most of the applications constituting the re- mainder of the work. §l. Definitions and Basic Properties; 3-radicals Let G be a finite solvable grOup. 1.1 Definition: A non-empty collection 3 of subgroups of G is called a Fitting set in g provided that the following three conditions are satisfied: i) N g H E 3 implies N E 3 ii) H,K e a, H,K 9 HK implies HR 6 3 iii) H e s implies Hx e s, for all x e G. A set of subgroups 3 of G for which i) holds is said to be normal subgroup closed. If 3 satisfies ii), it will be called normal product closed, and if iii) holds for 3, we will say that 3 is closed under inner automorphisms of G. We note that if 3 is normal subgroup closed and H E 3 then all subnormal subgroups of H belong to 3, i.e., 3 is actually subnormal Sub- group closed. If 3 is a Fitting set, the fact that 3 is not empty and normal Subgroup closed implies {l} E 3. By an 3-sub- group of G we mean merely a subgroup of G belonging to 3. 1.2 Examples: a) Let n be a set of primes. The collection Of all n-subgroups Of G is a Fitting set in G. b) The collection of all nilpotent Subgroups of G is a Fitting set in G. c) If N is a normal subgroup of G the collection Of all sub- groups Of N and the collection of all subnormal subgroups of N are Fitting sets in G. Let 3 be a Fitting set in G and H a subgroup of G. The restriction of 3 in H will be denoted by 3\H and is defined by 3]H = {S s H‘S E 3}. It is immediate that 3‘H is a Fitting set in H and if H g G that 3]“ = [s n H\s e a}. 1.3 Proposition: Let 3 be a Fitting set in G. Then: i) G has a unique maximal normal 3-subgroup G5 called the 3-radica1 of G. ii) G contains every subnormal 3-subgroup of G. 3 iii) N3 = N D Gy’ for every subnormal Subgroup N of G. Proof: i) Set G3 = n{H\H 9 G, H G 3}. ii) Let H 9 9 G with H G 3 and let H = H0 Q...9 Hk = G be a subnormal series joining H to G. For 0 Z i s k—l we have (Hi)? gHi 9Hi+1' Let x e Hi+1. Then (Hi); 9H): = Hi and . X X according to 1.1, 111), (Hi)3 E 3. Hence, (Hi)3 s (Hi)3 and we . X = obtain (Hi)3 (Hi)y' Thus (Hi)? 9 Hi+l' Therefore H = a? s (Hi)? s...s (H k)? = G5. iii) For N 9 9(3, ii) implies that N? s G? O N. Because G3 H N E 3 and G? H N 919, it follows that G:7 n N S N5. 1.4 Corollary: Let 3 be a Fitting set in G and H a normal subgroup of G. Then H5 9 G. Proof: This is the calculation in the proof of 1.3, ii). 1.5 Corollary: Let 3 be a Fitting set in G and A a minimal normal subgroup of G. Then either A E 3 or 3‘A = {1]. Proof: Suppose l # S E 3\A. Since A is abelian, 1 # S s A? Because A? 9(3 and A is minimal normal, it follows that A = A3 E 3. 1.6 Definition: Let S(G) denote the set of all subgroups of G. A function a: S(G) a S(G) will be called a radical function on 9 whenever i) a(H) g H, for all H E S(G) ii) 0,01") = O,(H)X, for all H e S(G) and all x c c iii) a(N) = N D a(H), for all H E S(G) and all N 9 H. Let a be a radical function on G and set 3a = [s e S(G)\a(S) = s}. If N a H 6 ya we have 01(N) = N n 9(H) = N n H = N so that N E 3a. Suppose H,K E 3d and H,K S HK. Then H = a(H) = H O a(HK) s a(HK) and K = a(K) = K H a(HK) s q(HK). Therefore, HK s a(HK) s HK and we obtain HK E 3a. Let H 6 3a and x e G. Then H = 01(H) and HX = get)" = 04H"). Thus, 3a is a Fitting set in G. Moreover, if H is any subgroup of G we have q(H) 2 a(H) fl H3 = G(Hg ) = Hy . On the other hand, a(H) 9 H and a(a(H)) = a(H) fl a(H)a= a(H)T So a(H) E 3a. Con— sequently, q(H) s H? and we obtain H3 = q(H). Conversely, If 3 is a Fitting :et in G then H a H? defines a function a? on S(G) satisfying the required conditions of definition 1.6. Furthermore, 3CY = {S E S(G)]ag(S) = S] = 3 {S E S(G)]Sy = S} = 3. Thus we have established 1.7 Proposition: Using the notation given above, a a 3g defines a bijective mapping of the set of radical functions on G onto the collection of Fitting sets in G. The inverse map is given by 3 H Q?- The Fitting sets described in 1.2 a), b) and c) are obtained by taking a(H) to be On(H)’ F(H) and N m H respectively, for all subgroups H of G. 1.8 Proposition: Let a1 and a2 be radical functions on G. Then the composite function a = a1 0 a2 is a radical function on G. Proof: Let H E S(G). We show that a satisfies i) - iii) of 1.6. i) Since a2(H) 9 H and q1(Q2(H)) = (a2(H))3 , it follows 0‘1 from 1.4 that a(H) = a1(a2(H)) = (a2(H))y g H. 01 ii) Let x E G. Since Q1 and a2 satisfy 1.6 ii), we have 01(Hx) = almzmxn = 011(012(H)x) = 011(a2(H))X = 01(H)X- iii) Let N 9 H. We first observe that a(H) = a1(Q2(H)) = al(H) fl a2(H) since a2(H) g H and a1 satisfies 1.6 iii). Thus, a(N) = a1(az(N)) = 01(a2(H) H N) = a1(H) fl [a2(H) D N] = [a1(H) fl a2(H)] = a(H) n N. Therefore a is a radical function. 1.9 Corollary: Let 31 and 32 be Fitting sets in G. Set 3 = 31 H 32 = {S‘S E 31 and S E 92}. Then 3 is a Fitting set in G and Hg = Hg n H? , for all H s G. l 2 Proof: Let a1 = a? and a2 = a? be the radical functions 1 2 corresponding to 31 and 32, respectively, under the correSpondence described in 1.7. According to 1.8, a = a 0 a2 is a radical 1 function on G. Hence, 3 = {S E S(G)\a(S) = S] is a Fitting set a in G. But, a(S) = S a 01(a2(S)) = S a a2(S) = S and a1(S) = S a 332 = S and 331 = 5 ¢ 3 E 31 H 32 = F. Therefore, 3a = 3 is a Fitting set in G. As in the proof of 1.8, we have Hg = H3 = a 0(H) = a1(a2(H)) = a1(H) fl 12(H) = H3 H H? = Hy D H? . a1 a2 1 2 §2. y-injectors 1.10 Definition: Let 3 be a Fitting set in G. A subgroup V of G is said to be Eamaximal in 9 if V E 5 and V s H S G with H E 9 implies V = H. V is called an S-injector pf g if V H N is g-maximal in N, for all N 9 9(3. The existence and conjugacy of y-injectors is now established for a Fitting set 3 in G. The proof given here is essentially that given by Fischer, Gaschutz and Hartley in [3]. Our proof requires the following proposition. 1.11 Proposition: Let 3 be a Fitting set in G and V an 9-maximal subgroup of G which contains the normal Subgroup A of G. Then i = {S/A‘A S S E 3} is a Fitting set in G/A and WA is E-maximal in G/A. Proof: We verify that g satisfies the three conditions of definition 1.1. i) N/A 9 H/A 6 § implies N a H e ‘3' which implies N e ti. Therefore, N/A E 5. ii) Let H/A, K/A a HK/A with H/A, K/A e 5. Then H,K e 31 and H,K 9 HR. Therefore HK E 3 and HK/A E 5. iii) H/A E 5 and x E G implies HX E 3 and consequently Hx/A e i That V/A is g-maximal in G/A is even more obvious. Proposition 1.11 allows a slight simplification in Hartley's proof of the following lemma. 1.12 Lemma (Hartley): Let 3 be a Fitting set in G. Let N 9 G and aSSume i) G/N is nilpotent ii) W is g-maximal in N 10 iii) W 5 V1 0 V where V and V are fi-maximal in G. 2’ 1 2 Then V1 and V2 are conjugate in G. Proof: Let G and 3 afford a counter-example with [G\ minimal. Since V1 0 N 4 Vi E 3, W s Vi H N and W is g-maximal in N, it follows that V1 0 N = W = V2 Q N. Let L = NG(W) and N* = N H L. Then V1 and V2 are fi—maximal in L and W is s-maximal in N*. Also, L/N* = L/N n L a LN/N s G/N so that L/N* is nilpotent. Thus, L < G implies V1 and V2 are con— jugate in L since G is a minimal counter—example. Therefore, W 9 G. Consider G/W. According to 1.11, g = {S/W‘W S S E 3} is a Fitting set in G/W, vl/w and VZ/W are E-maximal in cm, W/W is g-maximal in N/W and G/W/N/W is nilpotent. Hence W > 1 implies Vl/W and VZ/W are conjugate in G/W. Therefore, W = 1. Let M1 = NG(Vi) and let Ci be a Carter subgroup of Mi’ i = 1,2. Because Mi/Mi H N is nilpotent there is an integer r such that [M,,...,M,] s M_ H N. Therefore, [V_, M ,...,M j S l i l l l 1 Lw w r times r-l times V. H N = W = l, i.e., V, s 2 (M,) s C,. Let x E N (0,). Since 1 l m l l G 1 V.,V¥ 9 C. we have V.V¥ l l 1 l l E 3. Since Vi is g-maximal in Ci it x f 1 th t V. = V, _ N C = C = . ol ows a l l and x E M1 0 G( i) NMi( 1) Ci Hence Ci is self—normalizing in G and therefore is a Garter subgroup of G. But then there is an element y of G such that Ci = C . 2 y . . . y _ Thus, V1,V2 9 C2 and as before this implies V1 — V2. 1.13 Lemma: Let 3 be a Fitting set in G. Assume that every proper subgroup S of G has an fi-injector and that all 11 3-injectors of S are conjugate in S. Let M S G, G/M nilpotent, R an 3-injector of M and V g-maximal in G with R S V. Then V is an fi-injector of G. Proof: It suffices to show that H H V is an g-injector of H for every maximal normal subgroup H of G. Let N = H n M. Then G/N 1, the induction hypothesis implies V n is an S-injector of G Then V is 3-maxima1 in Gk-l k-l' G containing V no and G/Gk_1 is nilpotent. Lemma 1.13 k—l hmplies V is an F-injector of G. 1.17 Corollary: Let 3 be a Fitting set in G. If V is an F-injector of G and V S H S G, then V is an F-injector of H. Proof: Let 1 = G0 S...s Gk = G be a subnormal series with nil- potent factors and set Hi = Gi H H. Since 13 V H Hi = Ci 0 H n V = Cl H V is 9-maximal in Gi’ it is also F-maximal in Hi' 1.18 Definition: If K S H S G, then H/K is a factor of G. A subgroup V of G covers the factor H/K if (V H H)K = H. V avoids H/K if vnH=VnK. 1.19 Corollary: Let 9 be a Fitting set in G and V an H-injector of G. Then: i) V 0 H S L S H 9 S G and N S 9 L implies V n N and Vx n N are conjugate in , for all x E L. ii) V S L S G and H 9 L implies L = H NL(V H H). iii) V S L S G implies V covers or avoids each chief factor of L. Proof: i) V n H is an F—injector of H and by 1.17, V n H is an S-injector of L. Since N 9 9 L, V n N and VX m N are g-injectors of N. Thus, V 0 N and VX D N are g—injectors of and as such are conjugate in . ii) and iii) Let H/K be a chief factor of L and x E L. Then V H H and VX H H are conjugate by an element h belonging to . Thus,VflH=(VxflH)h=(VflH)Xh and xh E NG(V D H). Consequently, L = H NL(V H H). This implies (V n H)K S'L and therefore (V m H)K = H or (V n H)K = K. 1.20 Examples: a) Let N 9 G and let 3 be the Fitting set in G which con- sists of all subgroups of N. Then for any Subgroup H of G, 14 N H H is the unique H-injector of H. b) Let K S G and 3 a Fitting set in G. Let K = {sls S K} and set N'= 3 fl K1 According to 1.9, N’ is a Fitting set in G. If V is an y-injector of G, then V H K is an flfiinjector of G. For suppose (V H K) n N S W S N 9 S G with W E £2 Now W E 3' implies W S K and therefore W S N n K. But N n K S S G and V H (N D K) is y-maximal in N n K. Thus W = V n N n K and we have shown that (V n K) H N is QQmaximal in N for any N S S G. c) Let G be the symmetric group on 3 symbols. Let 31 be the Fitting set of all nilpotent subgroups of G and let 32 be the Fitting set in G consisting of all 2—subgroups of G. If A denotes the alternating group, then it is easy to see that A is the unique 31-injector of G and that the sylow 2—subgroups of G are 32-injectors of G. Because 31 0 32 = 3 the sylow 2, 2—subgroups of G are 31 fl 32-injectors of G. However, the intersection of A and any 32-injector of G is (1}. Thus, in spite of 1.9, it need not be the case that an 31 fl 32-injector of G be the intersection of an 31-injector and an 32—injector. §3. Fitting Classes By a class pf groups we mean a collection of solvable groups closed under the taking of isomorphisms. If 3 is a class of groups and G is a group, we will refer to the set of Subgroups of G which belong to 3 as the trace pf 2 ip G. 1.21 Definition: A Fitting class is a nonempty class of groups 3 Such that the trace of 3 in each group G is a Fitting set in G. 15 Because a class of groups is isomorphism closed it is immediate that a class of groups 3 is a Fitting class if and only if 5 is normal subgroup and normal product closed. Accord- ing to 1.14 and 1.15 we have 1.22 Theorem ([3], Satz 1): If 9 is a Fitting class, then each group G possesses 9-injectors and all g-injectors of G are conjugate in G. 1.23 Corollapy: A class of groups 3 is a Fitting class if and only if every group possesses S—injectors. A further fact concerning Fitting classes is recorded as 1.24 Proposition: Let 3 be a Fitting class. Define a set of primes n by: p E n a p\\G\ for some group G belonging to 3. Then 3 contains all nilpotent n-groups. A proof of 1.24 can be found in [4], Remark 1, p. 204. 1.25 Proposition: Let N’ be a Fitting set in G and X' a Fitting class. Then a = {s s (us/s” e K} is a Fitting set in c. Proof: We show that 3 satisfies the required conditions of definition 1.1. i) Let N s s e 3'. Then s/sy e K and we have N/Ny = N/N n s =- NSy/Sy s s/s” e x. This implies N/NN e K- N’ ii) Let A,B 9 AB and A,B E 3. Then A/Ay and B/By belong to K1 BecaUSe A/fiv = A/A n (A8)”,3 A(AB)N,/(AB)y and, in the same way, B/By,” B(AB)k/(AB)”0 we have AB/(AB)y 2 A(AB)~/(AB)V,' 3(AB)~J(AB)~ E x2 Hence AB E 3. 16 iii) If s e 54’ and x e G, then s/sye‘ sX/s; e 9(- 1.26 Examples: a) Let G be the full symmetric group on {1,2,3,4} and let V be the subgroup of G fixing 4. Let 3* be the Fitting class generated by V. According to 1.24, 3* con— tains all Z-groups and all 3—groups. Since V H 02(G) is not 3*-maximal in 02(G), V is not an 3*-injector of G. However, if 3 is the collection ofsubgroups of G consisting of all 3—sub— groups of G and the conjugates of V in G, then 3 is a Fitting set in G and V is an 3-injector of G. But, 3 cannot be the trace of any Fitting class in G since such a class would necessarily * contain 3 . b) Let R’ be a Fitting set in G and let fl denote the Fitting class of nilpotent groups. Then 3 = {S S G\S/%v E W} is a Fitting set in G and for any 3-injector V of G we have i) V” = G” and ii) V/GN' is an fl—injector of G/Gfl, Proof: According to 1.25, 3 is a Fitting set in G. Since fi': 3, we have QN'S 63' We first show that G?/%V = (G/Gy)n. Since G? E 3, Cg/(Gy)y' is nilpotent. But G” S (G?)y,9<3 so we obtain (G9)~,= 9%. Therefore, G3/9V is nilpotent. Let H/Gv be a normal nilpotent subgroup of G/Gy. Then H E 3 as HW'= av. This forces H S 63' Thus, Gy/Gy' is the maximal normal nilpotent subgroup of G/Gy. Since in any solvable group the Fitting subgroup contains its centralizer, we obtain CG(Gy/%V) S 63' Because 17 [Vfld 6?] S V~,fl G? = (G?) = G it follows that V S G . But W’ W, W’ 3 . V S G = 5 Gy S G? S V so that G” S VN’S G? Hence y ( y)g' G”, V” and VN'= qv. Therefore V/gv is nilpotent as V E 3. If V S w and W/Gy is nilpotent, then w e a which implies V = w. There— fore V/GN' is a nilpotent subgroup of G/GN' maximal with respect to containing (G/ According to [5], Satz 7.18, p. G3,.)72 = Gy/qv. 705, V/QV is an fl-injector of G/Gfl- §4. Fitting Sets and Direct Products. Let G and G be groups and set G = G x G . 1f 3 1 2 1 2 is a Fitting set in G, then 3 = 3‘ and 3 = 3\ are 1 G1 2 G2 Fitting sets in G1 and G2 respectively. Since Gi S G we have 31 = {S H Gi\S E 3]. In this section we investigate the reverse procedure, that is, given Fitting sets 31 and 32 in G1 and G2 respectively we construct a Fitting set 3 in G Such that 3‘G1 = $1 and 3\G2 = 32_ 1.27 Proposition: Let G1 and G2 be groups with Fitting sets 31 and 32 respectively. Let G = G X G 1 2 and let fii: G 4 Gi’ i = 1,2 denote the canonical projection of G onto Gi. Define 31 x 52 = {s s G\Hi(S) e 3i, i = 1,2} . Then 31 X 32 is a Fitting set in G and (31 X 32)\Gi = {S n Gi\s e 31 x 32} = a i = 1,2. i, Proof: i) Let N 98 E 3’1 x 32. Then ni(N) 9ni(S) 631, i = 1,2. This implies N E 31 X 32. 18 ii) Suppose H = AB, A,B 9 H and A,B E 31 X 32. Then him) = ni(A)niiG. 1 from the definition of 31 X 32- Thus, for each pair of Fitting sets 31, 32 in G1, G2 respectively, there is a Fitting set 3 of G1 X G2 such that 3‘61 = 31 and 3 G2 = 32. The next example shows that G1 X G2 may possess Fitting sets which cannot arise in this manner. , 2 , _ _ 1.28 Example: Let Gi = , 1 — 1,2. Let H — be the diagonal subgroup of G X G2 and put 3 = {H, [1]}. 1 Then 3 is a Fitting set in G1 X G2 but there do not exist Fitting sets 3i in Gi such that 3 = 31 X 32. If such Fitting sets did exist they would satisfy 31 = 3\G = {1}. But if i 31 = {1}, then "i(H) = E 3i. §5. Fitting Sets and Quotients An important aspect of the theory of Fitting sets in applications is the fact that it is quite often possible to trans- fer a problem in G to homomorphic images of G. This has been illustrated to a limited extent by the use of 1.11 in the proof of 1.12. In this section the quotient theory is further de- veloped. 19 1.29 Proposition: Let A be a normal subgroup of G. Let 3 be a Fitting set in G/A with 3-injector W/A and put 3 = {S S G‘SA/A E 3}. Then 3 is a Fitting set in G and W is an 3-injector of G. Proof: a) 3 is a Fitting set: i) Let N g s E 3. Then NA/A s SA/A E 5 which implies NA/A E 3 and therefore N E 3. ii) Let H,K E a with H,K a HR. Then HA/A, KA/A e 5 and HA/A, KA/A 9 HKA/A. Thus HKA/A e 3 and HK E 3. iii) If s e a and x e G, then SA/A e 3 which implies SxA/A E 3 and consequently Sx E 3. b) W is an 3-injector: Let V be an 3-injector of G. Since A E 3, A S V. It suffices to Show that V/A is an 3-injector of G/A; for then V/A and W/A are conjugate in G/A, Let N/A be subnormal in CIA and suppose v nN/A s L/A s N/A with L/A e 3?. Then L E a and V n N S L S N 9 S G. Since V is an 3-injector of C, it follows that V n N = L and that V/A is an 3-injector. 1.30 Proposition: Let 3 be a Fitting set in G and V an 3-injector of G. If A a c with A s V, then 5 = [S/A‘A s s e s} is a Fitting set in G/A and VIA is an 5- injector of G/A. Moreover, 3* = {T S GlTA E 3} is a Fitting set in G and V is an 3*-injector of G. Proof: That 3 is a Fitting set has been established in 1.11. To see that V/A is an 3-injector of G/A, suppose 20 VflN/AST/ASN/ASSG/A with T/AE3. Then VflNSTS N 9 S G and T E 3. Because V is an 3-injector of G, N n V = T. Therefore V/A is an 3-injector of G/A. Because 3* = {T S GiTA E 3] = {T S G‘TA/A E 3}, it follows from 1.29 that 3* is a Fitting set and V is an 3*— injector of G. Let 3 be a Fitting set in G, V an 3-injector of G and A S G contained in V. According to 1.30, there is a Fitting * set 3 of G containing all Subgroups of A and such that V * is an 3 -injector. The following examples indicate that in general * there is no consistent relation between 3 and 3 . 1.31 Examples: a) Let G be the symmetric group on three symbols and 3 the trace of the Fitting class of nilpotent groups in G. Let A be the alternating group. Then 3 = {S/A‘A S S E 3} = {1} and 3* = {T\TA E a} = {1,A) (:3. b) Let G be the symmetric group on four letters, A the alternating group and K the normal four subgroup of G. Let 3={S\SSSA} and 3={S/K\KSSE3}={1,A/K}. Then 3* = {T‘TK E 3] = {all subgroups of A} D 3. 1.32 Lgmmg: Let 3 be a Fitting set in G = HA where A S G, H E 3 and H H A is an 3-injector of A. Then H is an 3-injector of G. Proof: Let 1=G SG s...gc =A9G 9...SG =G bea 0 1 k k+l n chief series for G passing through A. According to 1.16, it 21 suffices to show that H 0 Ci is F-maximal in Gi for i = 0,1,... For i s k, this follows from the fact that H n A is an fi-injector of A. Suppose k < i s n and H m Gi < w s Gi with w E 3. Because G = HA it follows that G1 = (H n Gi)A and consequently w=(HnGi)(AnW)>HnGi. Butthen AflH for all x E G. A subgroup V of G such that V n N is pronormal in G for all normal subgroups N of G is called a strongly pronormal subgroup of G. 2.5 through 2.9 are all re5ults established in [2]. The parenthetical references pertain to [2]. 2.5 (Corollary 2.4): If V is normally embedded in G, then V is strongly pronormal in G. 2.6 (Corollary 2.5): If V is normally embedded in G, then V covers or avoids each chief factor of G. 2.7 (Theorem 2.6): If V is normally embedded in G and W is a subgroup of G which covers the chief factors of G that V covers and avoids the chief factors of G that V avoids, then W is conjugate to V in G. 25 2.8 (Theorem 2.11): Suppose V is normally embedded in G and V is nilpotent. If H/K is a p-chief factor of G, then NG(V) covers H/K if and only if V/CV(H/K) is a p-group. 2.9 (Proposition 4.2): If G has p-length S l and V is strongly pronormal in G, then V is p-normally embedded in G. By a p-maximal Subgroup of G we mean a maximal subgroup of G whose index in G is a power of the prime p. It is clear that a p-maximal subgroup of G is normally embedded a it is p-normally embedded. 2.10 Proposition: Let M be a p-maximal subgroup of G. M is normally embedded in G if and only if M/coreG(M) is a p'-group. Proof: Let Mp E Sy1p(M) and assume M is p-normally embedded. If coreG(M) = 1, G is primitive. Let A be the unique minimal normal subgroup of G. Then G = MA, M m A = l and A is a p- group. Let Gp be a sylow p—subgroup of G containing Mp. Since M is p-normally embedded, Mp 9 Gp and because Mp n A = l we have Mp S CG(A) = A. Therefore, Mp = l and M is a p'—group. If coreG(M) > 1, then M/coreG(M) is a p-maximal Subgroup of G/coreG(M) and according to 2.3, 2), is p-normally embedded in G/coreG(M). Since core )(M/coreG(M)) = 1, the argument G/coreG(M given above implies M/coreG(M) is a p'-group. Conversely, if M/coreG(M) is a p'-group, then M/coreG(M) is a Hall-subgroup of G/coreG(M). This implies M/coreG(M) is p-normally embedded in G/coreG(M) and hence M is p—normally embedded in G. 26 2.11 Theorem: Let p be a prime and G a group. Every p-maximal subgroup of G is normally embedded in G if and only if G has p-1ength s 1. Proof: (2): Let G be a minimal counter-example. Then every homomorphic image of G distinct from G has p-length s 1. It follows from [5], Hilfssatz 6.9, p. 693, that G is primitive possessing a unique minimal normal subgroup A with pl‘A‘. Let M be a complement for A in G. Then M is a core—free p- maximal Subgroup of G. According to the hypothesis M is normally embedded in G. By 2.11, M is a p'-group and it follows that G has p-length s l. «6: Let M be a p-maximal subgroup of a minimal counter— example G. Let K = coreG(M). If K # 1, then M/K is normally embedded in G/K which implies M is normally embedded in G. If K = l, G is primitive and because G has p-length s 1, the unique minimal normal subgroup of G is a sylow p-subgroup of G. Therefore, M is a p-complement and hence normally embedded in G. §2. Fischer Sets A Fischer class is a Fitting class 3 which satisfies: N S G E 3, N S T s G and \T/N\ a power of some prime = T E 3. It is known that if 3 is a Fischer class then the 3—injectors of any solvable group G are normally embedded in C (see [4], p. 197). We show that this is actually a consequence of the local theory. 27 2.12 Theorem: Let V be an 3-injector of G where 3 is a Fitting set in G satisfying: (*) N 9 S E 3, N s T s S and T/N a p-group for some prime p 3 T E 3- Then V is normally embedded in G. Proof: Let G be a minimal counter—example. Set K = coreG(V) and Suppose K # 1. Then 3 = {S/K‘K s s 6 5:} is a Fitting set in G/K and V/K is an 3- injector. If N/K g S/K e 3, N/K s T/K s S/K and T/K/N/K is a p-group, then N 9 s e 3:, N s T s s and T/N 2‘ T/K/N/K is a p-group. Therefore T E 3 and T/K E 3. Hence 3 satisfies (*) in G/K. By the minimality of G, V/K is normally embedded in G/K. Therefore coreG(V) = 1. Let A be a minimal normal subgroup of C. By the above remarks, A g 3. Hence 3 = {SA/A‘s e a] is a Fitting set in G/A with VA/A an 3-injector of G/A. Suppose N/A 9 SA/A e 5 where s e 3;, N/A 5 T/A s SA/A and T/A/N/A is a p-group. Then N = A(N n S), T = A(T n S). Since NflSgS,NnS sTnSss and Tns/N ns isap- group, (*) implies that T n S E 3 and consequently T/A = (T n S)A/A E 3. Therefore 3 satisfies (*) in G/A. This implies VA/A is normally embedded in G/A. Let p be the prime dividing the order of A. Then for all primes q # p, V is q-normally embedded in G. If Op'(G) # 1, we may take a minimal normal subgroup of p' order and repeat the 28 above argument to conclude that V is p-normally embedded in G. Thus 0 .(G) = 1. P Assume V < G where Vp E Sy1p(V). Then 3) G is a V G P and V 0 VC is an 3‘ G-injector of V . P V P G P . . . G Fitting set in Vp P Since 3‘ G clearly satisfies (*) in VG we obtain Vp normally G G V G embedded in V , i.e., V E Syl (V p). But V p = V . To see this, P P P P P P let x E G. Now V n v: and Vx H v: are 3-injectors of V G p, so there is an element y E v: such that (V n Vi)y = VX 0 Vi. Thus v:, v: s Vx n v: and VE, v: are sylow p-subgroups of VX n VG. Let z e VG such that vX = V”. Since yz 6 VC we p p p p 9 have V: s V:p 9 Vi. This shows that if Vp is normally embedded in V3, then Vp is normally embedded in G. Therefore v: = G. Now VA/A is p—normally embedded in G/A = Vi/A, so we have VpA E Sylp(G). Since V 0 Op(G) = (OP(G))5 = 1 we obtain Vp n A = 1. This implies A = Op(G) and Since Op,(G) = 1, it follows that A = CG(A). Put P = VpA. Since (*) holds in G, we have Vp E 3. According tol.3% Vp = P 9 P. Hence Vp S CG(A) = A. This con- 3 tradiction completes the proof. 2.13 Corollary: Let 3 be a Fischer class and V an 3-injector of a group G. Then V is normally embedded in G. 2.14 Theorem: Let 3 be a Fitting set in G and V an 3- injector of G. If there exists N 9 G with N s V and V/N nilpotent, then V is normally embedded in G. In particular, if V is nilpotent then V is normally embedded in G. 29 Proof: Let G be a minimal counter—example. Suppose N > 1. Then 3 = {S/N‘N s S S 3} is a Fitting set in G/N and V/N is a nilpotent subgroup of G/N which is an 3-injector. By the minimality of \G‘, V/N is normally embedded in G/N and consequently in G. Thus N = l and V is a nilpotent subgroup of G. If fl denotes the Fitting set of nilpotent subgroups of G, then X'= 3 n W is a Fitting set in G and V is a xeinjector of G. Since every subgroup of a nilpotent group is subnormal, X’ is a subgroup closed Fitting set and as such satisfies (*) of 2.12. Remark. The hypothesis in 2.14 concerning the existence of Such a normal subgroup N is equivalent to requiring that V/G? be nilpotent. 2.15 Corollapy: Let 3 be a Fitting set in G, V an 3-injector of G and N a normal subgroup of G such that V H N/coreG(V 0 N) is nilpotent. Then V n N is normally embedded in G. Proof: It follows frOm 1.20 b), that 3‘N is a Fitting set in G and V n N is an 3\N-injector of G. Since coreG(V n N) = 63‘ , N theoren12.14imp1ies V n N is normally embedded in G. Combining 2.7 with 2.14 yields a generalization of Theorem 4.4 of [2]. 2.16 Theorem: Let 3 be a Fitting set in G and V an 3-injector of G. If V is nilpotent then NG(V) covers the p-chief factor 3O H/K of G if and only if V/CV(H/K) is a p-group. Let 3 be a Fitting class. An 3-Subgroup V of G is called a Fischer 3-subgroup of G if V contains every 3-subgroup of G which it normalizes. That the 3~injectors of G are Fischer 3-subgroups follows from 1.17. Fischer and also Hartley have shown that if 3 is a Fischer class, then the Fischer 3-subgroups are precisely the 3-injectors of C (see [4], Theorem 1). Again, this is a local reSult and follows immediately from 2.12 and 2.17 Theorem: Let 3 be a Fitting set in G such that for each S s G the 3-injectors of S are normally embedded in S. Then the following are equivalent: 1) W is an 3-injector of G. ii) W E 3 and W contains each 3-subgroup of G which it normalizes. Proof: That i) implies ii) is clear. For the reverse implication, let G be a minimal counter- example. Suppose 6% > 1. Then 3 = {S/Gy\G? S S E 3} is a Fitting set in G/Gg, subgroups of G/G? have normally embedded 3-injectors and W/Gy satisfies ii). By the minimality of G as a counter- example W/Gy is an 3-injector of G/Gg' However, this implies W is an 3-injector of G. Consequently, G3 = 1. Let A be a minimal normal subgroup of G. Then A E 3 and E = {SA/A‘s e a} is a Fitting set in G/A such that for A s T s G we have VTA/A is an 3-injector of T/A where VT denotes 31 an 3-injector of T. Since VT is normally embedded in T, VTA/A is normally embedded in T/A. Now if WA/A satisfies ii) in G/A it would follow that WA = VA for some 3-injector V of G and then that W is an 3-injector of G. Thus there exists an x e a with WA/A s N A(XA/A) but such that G/ XA i WA. If NG(XA) < G, then W is an 3-injector of NG(XA) which implies W n XA is an 3-injector of XA. Since X is an 3-injector of XA, some conjugate of X in XA is W n XA, i.e., Xy = W n XA, y E A. But then XA = XyA 5 WA. Therefore XA 9 G. An almost identical argument now shows that G = WXA. Because XA 9 G we have G = AXNG(V fl XA) = AXNG(X) = ANG(X), A 0 NG(X) = 1 for a suitable 3-injector V of G. Suppose A s K < XA with XA/K a chief factor of G. Then X n K is an 3-injector of K and K = A(K 0 X). Since WA/A s N (K/A) and \K n x\ < {x}, induction on \x\ implies G/A K 3 WA. But this means W m K is an 3-injector of K so that W n K = V n K for an 3-injector V of G. This forces V,W S NG(W n K) and consequently G = NG(W m K). Because G3 = l we have W 0 K = l and therefore X n K = 1. This implies K = A and hence XA/A is a chief factor of G. Let B/A be a minimal normal subgroup of G/A and assume B # XA. Let H = XAB = XA(W n H) and put K = (w n H)A. Then K/A g G/A since H/A is abelian and G = Hw. Now w n H = w n K is an 3-injector of K so W n K = V H K and V,W s NG(W n K), where V is an 3-injector of G. But V n K cannot be normal in G since 1 < W H H S NG(V n K) and G3 = 1. Therefore 32 B = XA and XA/A is the unique minimal normal subgroup. Let c = cG(A). Then 0 SG and CIA = l or XA/A s C/A. If XA S C, then A S NG(X) which is not possible. Thus, C = A. Let p be the prime dividing {A\ and Wp E Sy1p(W). Since W is an 3-injector of WA we have Wp E Sy1p(W:A). Then 1 = Wp n A is a sylow p—subgroup of the p-group WZA n A. Hence, wWA n A = 1 and WWA s c (A) = A. Therefore, WWA = l and p p G D (\W\, \A\) = 1. But then W and WA 0 NG(X) are Hall p'—subgroups of WA and as such are conjugate. Thus for some x E G we have Wx S NG(X). Because there is an 3—injector V of G with X 9 V S NG(X) we obtain the contradiction: V and W are conjugate. 2.18 Corollary: (Fischer's Theorem). If 3 is a Fischer class then the Fischer 3-Subgroups are 3-injectors. §3. More On Normally Embedded Subgroups. Let V be a normally embedded subgroup of G. In this section it is shown that there is a very simple Fitting set 3 in G such that V is an 3-injector of G. In fact, the 3- injectors of any subgroup H of G are the intersections of H with those conjugates of V which reduce the same sylow systems of G that H reduces. By a sylow system (also called a sylow basis) of G we shall mean a complete set of pairwise permutable sylow subgroups of G, exactly one for each prime dividing \G‘. If 2 = {Tp\p\\G\} is a sylow system of G and H S G, we say that H reduces 2 if H H g = {H 0 Tp\p|\G\} is a sylow system of H. my 33 For results concerning the basic prOpertieS of sylow systems, the reader is referred to [5], Chapter VI, in particular paragraphs 2 and 15. 2.19 Theorem: Let V be normally embedded in G. Then 3 = {SX\S S V, x E G] is a Fitting set in G. Moreover, if H is any subgroup of G and Z is a sylow system of G which H reduces then Vy n H is an 3-injector of H where Vy is a con— jugate of V which reduces 2- Proof: It is enough to Show that if A S V and B S VX with A,B SAB, then AB SVy for some y E G. If V is a normally embedded p-group, then V is a sylow p-subgroup of the normal subgroup VC and 3 is the intersection of the Fitting sets 31 = {S‘S S VG} and 32 = {S s G‘S is a p- group}. Therefore if V is a p-group, 3 is a Fitting set. Now let V be any normally embedded subgroup of G. Let 2 be a sylow system reducing into H = AB and let Vy be a conjugate of V reducing 2- For each prime p let Tp be the P A = A n T and B = B D T . Since A and B are normal Sub- P P P P sylow p—subgroup of G belonging to 2, HP = H 0 T , V: = Vy n T i P groups of H, A and B also reduce 2 so that Apo = Hp for all rimes . Hence H = H H = H A B . p p p p p p p Let 3p = {321$ S V1, 2 E G}. By the above remarks 3p is a Fitting set and V: is the 3p-injector of Tp. Since A ,B E a and A ,B s T we have A B s (T ) = vy. There- p p p p p p p p p 3 p fore H = H H = n A B S H Vy = Vy. This proves 3 is a Fitting p p p p P p p set in G. W ‘ 34 Let H S G and assume V and H both reduce the sylow system 2 = {Tp\p\\G\]. To complete the proof it is enough to show that V n H is 9-maximal in H. Suppose V n H S W S H with W E 3. Because of the nature of 9 we may assume W = Vg n H for some g E G. Let p be any prime. Set Vp = V D Tp E Sylp(V). Since = V: fl‘Tp. Because H reduces E, p G G H= T= H=T H VanSH andvpn vanp VpnTpfl (pn)n V is normally embedded, V (v: H H), it follows that Vp H H E Sylp(V: n H). Since V and H both reduce 2, so does V 0 H and Vp n H = V D H n Tp E Syl (v n H) holds. Let R e Syl (vg n H) such that v n H s R . P P P P P Now any sylow p-subgroup of Vg is a sylow p-subgroup of V2 so G we may choose Q E Sy1p(Vg) such that Vp H H S R S Q S Vp. Hence p G G V H S R S H S V H. As V H H S 1 V H , 't follows p H p Q n p 0 p E y p( p H ) 1 that \v n H‘ = \Vg n H\ = m. 2.20 Corollary: Let V be normally embedded in G. Then i) V n H S L S H 9 9 G and N 9 S L implies V n N and VX n N are conjugate in for all x E L. ii) V S L S G implies V covers or avoids each chief factor of L. iii) V S L S G and H S L implies L = HNL(V 0 H). Proof: See 1.19. 2.21 Corollagy: Let V be normally embedded in G. If x E G, S s VX and V normalizes S, then S S V H VX. ‘V’ 35 Combining 2.19 and 2.14 necessary and sufficient conditions in order that a subgroup be normally embedded are obtained. 2.22 Theorem: Let V be a subgroup of G. The following are equivalent: i) V is normally embedded in G. ii) {SX\S S V, x E G} is closed under the taking of normal products in G. iii) For each prime p and sylow p-subgroup Vp of V, {SX\S S VP, x E G] is normal product closed in G. iv) Given any sylow subgroup R of V, there is a Fitting set 3 in G such that R is an S-injector of G. Proof: (1) =2 (ii) is 2.19. (ii) :2 (iii): Let 31 = {Sx\s s v, x e G} and let 6p be the Fitting set of all p-subgroups of G. Then because 3 is a Fitting set, so is 3 fl Sp = {SX\S S VP, x E G] and therefore is normal product closed. (iii) = (iv): Let R be a sylow p-subgroup of V and 3p = {SX\S S R, x E G}. According to the hypothesis SP is a Fitting set in G. But R is clearly an gp-injector of G. (iv) = (i): V is normally embedded if its sylow sub- groups are normally embedded. Let Vp E Sylp(V). Then Vp is an F-injector of G for some Fitting set 3 in G. According to theorem 2.14, Vp is normally embedded in G. Let V be a normally embedded subgroup of G. Then there is a Fitting set 3 = {SX\S S V, x E G} such that every 36 g-maximal subgroup of G is an #—injector. A partial converse is also true. 2.23 Theorem: Let 9 be a Fitting set in G. If every 3-maximal subgroup of G is an fi-injector of G and every sylow subgroup of an y-injector belongs to 9, then the g-injectors are normally embedded in G. Proof: Let V be an 5-injector of G and VPE SylP(V). Let P E Sy1p(G) such that P n V = VP. Consider Py' By the hypothesis Vp S Pg. But we also have that P? is contained in some 3- injector of G. Thus, Vp = P3' But P:7 is normally embedded in G. To verify this it is enough to show that {SX\S S Pw’ x E G} is normal product closed in G. Suppose A S Py’ B S P; and A,B 9 AB. Then AB is a p-group so AB S Py for some y E G. Since A,B are also elements of 3, y _y A — . BSO”)? P3 Remarks: 1) The condition that every g-maximal subgroup is an fi-injector is equivalent to 3 S {SX\S S V, x E G} where V is an S-injector of G. 2) The requirement that sylow subgroups of y-injectors be- long to 3 is necessary as the example given in 1.26 a) shows. 3) Let 3 be a Fitting class and let n be the set of all primes such that G E 3 implies \G\ is a n-number. For each p E n, 3 contains all p-groups (see 1.24). Hence for any group G, the sylow subgroups of the S-injectors of G belong to 3. In fact, the sylow p-subgroups of G belong to 3 for each may 37 p E n. Suppose that every 9-maxima1 subgroup of G is an #— injector of G. Then for each p E n and P E Sylp(G) we have P E 3 and so is contained in some fi—injector of G. Therefore the 3-injectors are precisely the n—Hall subgroups of G. Referring again to Chamber's work, we obtain 2.24 Theorem: Let G have p-length S l for all primes p and let V be a subgroup of G. There is a Fitting set 3 in G such that V is an 5-injector if and only if V is strongly pronormal in G. Proof: This follows from 2.19 and 2.9. 2.25 Example: (Rose [6], Example 2.5). Let G be the wreath product of a dihedral group of order 8 by a cyclic group of order 3. Then C may be expressed as G = HQ where 4 2 _ 2 _ H — X X , Q - and x1 ~ yi - (Xiyi) — 1» u = 1, x: = xi+1, y: = yi+1 for all i (subscripts are inter- preted modulo 3). Then H and Q define a sylow system 2 of G. Let D = NG(Z). If D2 denotes the sylow 2-subgroup of D, then D2 is precisely the diagonal subgroup and is subnormal but not normal in G. Therefore D2 cannot be pronormal in G and consequently D is not normally embedded. Since G has nilpotent length 2, D is also 3 Carter subgroup of G and hence is pro- normal in G. This example shows that "strongly pronormal" cannot be replaced by pronormal in 2.24. 38 2.26 Corollary: Let 3 be a Fitting set in G and assume LP(G) S 1 for all primes p. Then a subgroup V of G is an g-injector of G if and only if V E 3 and V contains each 3-subgroup of G which it normalizes. Proof: Since any subgroup S of G has p—length S l for all p, the fi-injectors of S are normally embedded so the hypotheses of 2.17 are satisfied. §4. Normal Fitting Sets 2.27 Definition: A Fitting set 3 in G will be called normal if the g-injectors of each subgrOup H of G are normal in H. A Fitting class 3 is called normal if the trace of 3 in each group G is a normal Fitting set in G. We remark that if 9 is a normal Fitting set in G and H S G, then H? is the unique S—injector of H. Our goal in this section is a proof based on the theory of Fitting sets of the following theorem due to Blessenohl and Gaschutz: 2.28 Theorem: ([1], Satz 6.2). Let 6 be a family of normal Fitting classes each # {1}. Then fl{§13 E 6} is a normal Fitting class # 1. In particular, there is a smallest normal Fitting class # {1}. Our proof of 2.28 is based on the following result con— cerning normal Fitting sets. 39 2.29 Theorem: Let 31 and 32 be normal Fitting sets in G. Then 31 D 32 is a normal Fitting set in G. Proof: According to 1.9, 31 H 32 is a Fitting set in G. We take G to be a counter-example of minimum order and set 3 = 31 H 32. Then every proper subgroup of G possesses a normal 3-injector. Let Vi be an 31-injector of G, i = 1,2 and let V be an 3-injector of G. Let A = V1 0 V2. Then A is the 32-injector of V1 and the 31-injector of V2. Hence A = V n V1 = V 0 V2 = V1 n V2 = G31 m 052 = G3 = coreG(V). Suvpose A aé 1. Set 51 = {S/A‘A s s e 311], i = 1.2. Then 31, 32 are normal Fitting sets in G/A with injectors Vl/A, V2/A respectively. Since \G/A\ < \G\, 31 H 32 is a normal Fitting set in G/A. However, it is clear that V/A is an 31 fl 32-injector of G/A and therefore V S G. Thus, A = V1 n V2 = V H V1 = V m V2 = 1. Let L = VVIVZ. If L < G, then 3 are normal 1‘1.’ 3211. Fitting sets in L and it follows that 0 32\L = 3‘L is ”ML a normal Fitting set in L. This implies l = V1 0 V2 = V. Hence G = VV1V2' Let R = V V and Suppose R < G. Then 3 1 and 3 liR 21R are normal Fitting sets in R, V is the 31lR-injector of R l and V is the 3 -injector of R. Because V is also the ZiR 3‘R = 31\R fl 32\R-1n3ector of R we obtain 1 = V1 0 V = V. Therefore G = V1V2. Similarly G = VZV' Now suppose V1V2 < G. Then V1 0 V2 = l is the 31 n 32- 1nJector of V1V2. But V1V2 9 C so V H V1V2 is the 40 31 fl 32—injector of vlvz. Thus V n V1V2 = 1. This yields iVHV1V2\ . 1 \ ] = Ffjfidffizgl = \V\\V1V2\ = \V\\V1\\V2\. Since we also have , \Gl = [V\\V1\ it follows that V2 = l, a contradiction. There- G=VV1=W =VV and VnV1=VflV =V flV=l and 2 l 2 2 l 2 \V\ = PM = \V2\ - Suppose V1 is a maximal normal Subgroup of G, say G , ; \G:V1\ = p. Then p = \szll = iViT = \V2\ = \Vll' But if G is a p-group every injector is normal. Hence V cannot be a 1 maximal subgroup of G. Let H be a maximal normal Subgroup of G containing V If V n H = l we obtain H = H D V1V = V1(H n V) = V1. 1. Therefore vnua‘l. But an= =hncy=1 since H<)G H 3 and V is an 3-injector. This final Contradiction completes the proof of 2.29. For our proof of 2.28 we also require 2.30 Proposition (J. Cossey): Every normal Fitting class # 1 contains the Fitting class of nilpotent groups. This is Satz 5.1 of [1]. Proof of 2.28: Let X'= fl{3\3 E 6} and let G be a group. Then each 3 E 6 yields a normal Fitting set in G which contains all the nilpotent subgroups of C. By 2.28, the intersection of these is a nomral Fitting set in G containing all nilpotent Subgroups of G. Since this intersection is clearly the trace of X’ in G we obtain the theorem. CHAPTER III INJECTORS §1. Cover-Avoidance Characterizations of Injectors. Let 3 be a Fitting set in G and V an 3-injector of G. As pointed out by Chambers ([2], Theorem 4.1) the 3—injectors of G are characterized by their cover avoidance property as long as V is normally embedded. Thus, if the 3—injectors of G are normally embedded we have: A subgroup W of G is an 3—injector of G if and only if W covers every chief factor of G which V covers and avoids every chief factor of G that V avoids. We give some results concerning the general case. 3.1 Theorem: Let 3 be a Fitting set in G and V an 3-injector of G. A subgroup W of G is an 3-injector of G if and only if W E 3 and V and W cover precisely the same chief factors of G. Proof: Necessity is clear. To prove the conditions suffice we proceed by induction on \G\. Let A be a minimal normal subgroup of G. According to 1.30 and 1.33 there is a Fitting set 3 in G/A such that VA/A is an 3-injector of G/A. Since V and W cover precisely the same chief factors of G we have A S V a A S W so that WA/A E 3. 41 42 Let H/A/K/A be a chief factor of G/A. Then H/K is a chief factor of G and VA n‘H = A(V FIH). Thus (VA 0 H)K = H a (v n H)K = H, i.e., VA/A covers H/A/K/A a V covers H/K. In the same way WA/A covers H/A/K/A w'W covers H/K. Hence WA/A covers precisely the same chief factors of G/A as the geinjector VA/A of G/A. By induction, WA/A is an 3-injector of G/A and we may assume WA = VA. If A E 3 we obtain V = W. If A é 3, 1.32 implies W is an 3-injector of VA. Since V is also an 3-injector of VA we have V and W are conjugate. 3.2 Corollagy: Let 3 be a Fitting set in G and V an 3-injector of G. Assume that either W 6 3 or that V is normally embedded in G. Then the following are equivalent: i) W is an 3-injector ii) Every sylow subgroup of W is a sylow subgroup of some 3-injector of G. Proof: That i) 2 ii) is trivial ii) = i): Let H/K be a p-chief factor of G. Then V covers H/K a any sylow p-subgroup Vp of V covers H/K a any conjugate of Vp covers H/K a W covers H/K. Thus W and V cover precisely the same chief factors of C. If V is normally embedded 2.7 implies W is an 3-injector and if W E 3 the same conclusion follows from 3.1. By strengthening the cover-avoidance requirement we need no longer require that W belong to 3. 43 3.3 Theorem: Let 3 be a Fitting set in G and W S G. The following are equivalent: i) W is an 3-injector of G ii) for each 3-injector V of G we have = L S G implies V and W cover precisely the same chief factors of L. Proof: To show that ii) implies i) we proceed by induction on [G]. Let A be a minimal normal subgroup of G. As in the proof of 3.1, for an 3-injector V of G we have VA/A is an 3-injector of G/A and if s L/A s G/A, the fact that ii) holds in G implies that VA/A and WA/A cover precisely the same chief factors of L/A. Thus ii) holds for WA/A and 3 in G/A. By induction, WA/A is an 3-injector of G/A. Hence for some 3-injector V of G we have WA = VA. If A E 3 we have W = V and we are finished. Therefore we may assume A 6 3 and coreG(V) = coreG(W) = 1. If VA < C, We have W and 3lvA satisfy ii) in VA and the induction hypothesis implies V and W are conjugate. Thus we assume VA = WA = G. But then V and W are core-free maximal Subgroups of G and it follows from [5], Satz 3.8, p. 165 that V and W are conjugate. That i) implies ii) follows by 1.17. §2. Injector Subgroups. A subgroup V of G will be called an injector subgroup of G is there exists a Fitting set 3 in G such that V is an 3-injector of G. We denote the collection of all injector subgroups of G by Inj(G). 44 3.4 Definition: Let U S G. We define a collection of subgroups G) 3U of G as follows: 3U = #:03k. where 30 = who 6 G). {s s G\s s L e ak_1}, if k is odd 3 = , k = 1,2,... [AB\A,B 9 AB, A,B E 3 if k is even k-l}’ 3.5 Proposition: Let U S G. Then 3U is a Fitting set in G and 3U = rms‘s is a Fitting set in G with U e s}. Proof: If 3U is a Fitting set, it is clearly the smallest Fitting set in G containing U so must be FW3‘U E 3, 3 a Fitting set in G}. Suppose S E 3U and x E G. Then S E 3k for some integer k. If k = 0, then sx also belongs to so and hence to 3U. Assume W E 3j implies Wx E 3j for all subgroups W of G and integers j< k. If k is even, 8 = AB with A,B SS and x x x x x A,B E 3k_1. But then A ,B E 3k_1 and S — A B E 3k. If k is odd, 8 SL E ilk-1 which means Sx SLX E 31k_1 so that x x S E 3k. Thus S E 3U. Let N SIS E 3U. Then S E 3k for some k and therefore N E 3 Hence N E 3U. k+2' Finally, if A,B E 3U with A,B 9 AB, then there are integers i and j with A E 31, B E 3j. Thus A,B E 3L where L = max {i,j} and consequently AB E 3£+2 ; 3U. This shows that 3U is a Fitting set in G. 45 3.6 Proposition: Let V be a subgroup of G. V is an injector of G if and only if it is an 3V-injector of G. Proof: Sufficiency is clear. Suppose V is an 3-injector for some Fitting set 3 in G. Then 3V S.3 and V is an 3V n 3 = 3V-injector of G. 3.7 Proposition: Let V be a subgroup of G. V is an injector of G if and only if V n'N contains every 3v-subgroup of N which it normalizes for all 'N S 9(3. Proof: Suppose V is an injector of G. Then V is an 3v-injector of G. Let N99G,KSN,L=NN(K),VflNSL and KE3V. Since V (IN is an 3V-injector of L, we have K S'L S'V F\N. 3v Conversely, let N 5) 9G. Then V nN SW SN and W E 3 V implies V FIN = W. Hence V FIN is 3V-maximal in N. Therefore V is an 3V-injector of G. In general, 3V appears to be quite difficult to determine. However, we have seen that if V is normally embedded av s {sx\s s v, x e G} and that if V is nilpotent then 3v = {sx\s s v, x e G} s'v e Inj(G). According to 2.19 all normally embedded subgroups of G belong to Inj(G). If Lp(G) S l for all primes p, then 2.24 and 2.9 imply that Inj(G) is precisely the set of normally embedded subgroups of G. Example 1.20 b) shows that V E Inj(G) implies {V r1N\N s G} ; Inj(G). This is clarified by the following proposition. 46 3.8 Proposition: Let N be a normal subgroup of G and V E Inj(N). Then V E Inj(G) «TV is pronormal in G. Proof: For any subgroup S of G containing V, let 3V(S) denote the Fitting set generated by V in 8. Suppose V is pronormal in G. We show 3V(N) = 3V(G). Since N S C it is clear from 3.4 that equality can fail if and only if for some x E G, V and VX are not conjugate in N. But, as V is pronormal in G, V and VX are conjugate in S N. Thus V is an injector of G. The reverse implication is clear. 3.9 Example: Let G = 34‘L 02 where (32 = is cyclic of order 2. Let V E Syl3(S Then V X V E Sy13(G), V X l = 4). X . ' VXVfl(A4 l) and A4X19A4XA49G Thus VXl is normally embedded in A4 X A4 and is therefore an injector of that normal subgroup of G. However, V X l is not an injector of G since (V X l)X = 1 X V and V X l, 1 X V are normal subgroups of = V X V. 3.10 Proposition: Let VEInj(G) and N 9G. Then VN/N E Inj(G/N). Proof: Induction on \G‘. Let A be a minimal normal subgroup of G contained in N. From 1.30 and 1.33 it follows that VA/A is an injector subgroup of G/A. By induction, VN/A/N/A is an injector subgroup of G/A/N/A. Since an isomorphism will clearly carry a Fitting set to a Fitting set, we obtain VN/N E Inj(G/N) using the natural isomorphism from G/A/N/A onto G/N. 47 3.11 Proposition: Let V E Inj(G) and N SIG» Then VN E Inj(G). Proof: According to 3.10, VN/N is an injector of G/N and 1.29 implies VN is an injector of G. We conclude with a result concerning certain permutable injectors. Professor Yen has shown that the collection of normally embedded subgroups of G reducing the same sylow system of G forms a lattice of permutable subgroups ([7], Theorem 2). Since the collection of normally embedded subgroups of G is contained in Inj(G), it is natural to ask the extent to which Yen's theorem holds for injector subgroups of G. At this time we have only succeeded in establishing the following: 3.12 Theorem: Let V and W be permutable injectors of G with (\V‘, \W‘) = 1. Then VW is an injector subgroup of G. Proof: Let n1, respectively n be the set of prime divisors 2 of V, reSpectively W. Then 3V C 8% and 3w C(S . Define n 3 = {AB‘A E 3 , B E 3 , AB = BA}. We :how that 3 i: a Fitting set in G. i): Let N SJAB = L E 3 where A E 3V, B E 3w. Then A and B are Hall-subgroups of L. Since N 9L, A D N and B H N are Hall-subgroups of N, and it follows that \N\ = [A F\N\\B F1N‘. Thus, N = (AnN)(B nN) es. ii): Let H,K QHK where H = AlBl E 3 and K = A232 E 3. We may assume that H, K, A1, B1, A2, B2 all reduce the same sylow system 2 of G. Because A1, A2, B1, B2 are all normally 48 embedded in HK, Yen's theorem quoted above implies they are pair- w13e permutable. Hence, HK = AlBlAZBZ = A1A2B1B2. Let U be an 3V-injector of HK reducing 2- Then U is contained in a n1- Hall subgroup of HK and U 0 H = A U fl'K = A2, since A A l’ 1’ 2 are 3V-injectors of H, K, respectively, which reduce 2. Because A . _ ' = - . A1 2 IS a n1 Hall subgroup of HK, we obtain A1A2 U E 3V Similarly, B1B2 E 3w. Therefore, HK E 3. iii): That 3 is closed under inner automorphisms of G is clear. It remains to show that VW is an 3-injector of G. Let NSQG and suppose VWflNSTSN where TE3. Then T=AB with A E 3 , B E 3w. Now V FIN is a nl-group, W FIN is a nz-group, A is a n -Ha11 subgroup of T and B is a nZ-Hall sub- 1 group of T, so there exist x,y E T such that V FIN S Ax E 3V and w n N s 13X 6 aw. But, T = AB = AxBy ([5],Hi1fssatz 4.5, p. 675). Thus we assume V H N S A and W O N S B. Since V n N and W FIN are 3V and 3w-injectors, respectively, of N, it follows that V FIN = A and W fl'N = B. Thus, VW FIN = AB = T and VW FIN is 3-maximal in N. This proves VW is an 3—injector of G and completes the proof of 3.12. BIBLIOGRAPHY w. ... [1] [2] [3] [4] [5] [6] [7] BIBLIOGRAPHY Blessenohl, D. and Gaschutz, W., Uber normal Schunck- and Fittingklassen, Math. Zietschr. 118 (1970), 1-8. Chambers, G.A., péNormally Embedded Subgroups of Finite Solvable Groups, J. of Algebra 16 (1970), 442-445. Fischer, B., Gaschutz, W. and Hartley, B., Injectoren endlicher aulesbarer Gruppen, Math. Zietschr. 102 (1967), 337-339. Hartley, B., On Fischer's Dualization of Formation Theory, Proc. London Math. Soc. (3) 19 (1969), 193-207. Huppert, B., Endiche Gruppen I, Die Grundlehren der math. Wissenschaften, Band 134, Springer4Verlag, Berlin, Heidelberg and New York, 1967. Rose, John 8., Finite Soluble Groups With Pronormal Systems Normalizers, Proc. London Math. Soc. (3) 17 (1967), 447-469. Yen, Ti, Permutable Pronormal Subgroups, Proc. AMS, 34 (1972), 340-341. 49 RRRRRR M.WWWEIIIAIIIIIZIIIIIIIQIIIII 2