CTT-PARTETEONS 0F FINIR'E GROUPS Thesis fo: the Degree of Ph. D. MtCHIGM STATE UNWERSEW JAMES W. RECHARBS 196.8 E 4mlfltSnflfil'fir M" 4" I 'fl - 1‘!- ” "! h I Q. ‘l. . . I gay 4: F .‘= 3 V) 29’ ' .2 a. » «a. .5 I ‘1 Q F" ‘ u V1.1. ”We -. . «mmfiv This is to certifg that the thesis entitled 3 11 - Partitions of Finite Groups presented by James W. Richard: has been accepted towards fulfillment of the requirements for flL-L degree inmic‘ '7- ‘1 F‘M/ /! \ :7 .' : ,1 X {/1 it I - [’7 I - .- ‘ 042% :4 ”7,11,, Major professor 0-169 ABSTRACT Cfl-PARTITIONS OF FINITE GROUPS by James W3 Richards Often in the study of mathematical systems, information about a given system can be obtained from presupposed properties other than the intrinsic ones. More specifically, in the study of finite groups-~with which we shallbe concerned-~information about a given abstract group sometimes can be extracted if one presumes certain conditions on its lattice structure. Reinhold Baer [1] has studied finite groups which admit a partition, that is, a family 0 of subgroups of G .whichcover G so that distinct numbers have trivial intersection. The purpose of this dissertation is to consider more general types of partitions. Let W [ n] denote the set of prime divisors of the positive integer n. If 11 is a given set of primes, then a. subgroup H of G is said to be a n-subgroup if n[ 1Hl] 3n and a rfi-subgroup if n [\HH fin = (25. Similarly, an element g of G is said to be a W-element if 11 [lg‘JETT and a TT'se’lement if TT[ ‘gn fin = ¢. If G is a finite group and if “SUI: ‘G‘ J, then we define a cTT-partition of G as follows: a family 0' of subgroups of G which cover G such that no member is a proper subgroup of any other and distinct JAMES W. RICHARDS members intersect as a cyclic TT-subgroup. If U: {p}, then we call 0 a cp-partition of G. Also, 0 is said to be trivial if U = {G}. In chapter I, we prove the following results: If 0' is a CV -partition of a finite group and if the center Z(G) of G contains a nielement of composite order, then 0 is trivial. If G is a finite nilpotent group and if TTC TV I: lGI J , then G admits a non-trivial CTT -partition iff G is the direct product of a cyclic TT-subgroup with a Sylow q-subgroup Sq of G where q g 11. ISq‘ 15 q, and Sq is not generated by all those elements of Sq which do not have order q. If 0 is anon-trivial cTT-partition of G such that TTflTT [[G:h(G) J] = 03, then there is a normal Hall n-subgroup H of G such that T = {L/HzLEG} is a Baer partition of G/H. In chapter II, we define "(J-admissibility" and derive the following results: If 0' is a non-trivial cp—partition of a finite group G such that all components are normal, then G is a p-group' or the semi-direct product of a cyclic p-group by an JAM ES W. RICHARDS elementary abelian q-group where q J5 p. If 0' is a normal non-trivial cp-partition of a finite group G and if L is a component of O' which is not self-normalizing, then there is a prime divisor q of I: NG(L):LJ such that L is an Hnugroup where n divides q- exp (0'). In chapter 111, we define a Frobenius cp-partition of G as a normal cp-partition which contains a prOper self-normalizing . component. We first prove the following result which is a partial generalization of a well-known theorem of Frobenius: If H is a self-normalizing Hall subgroup of a finite group G such that g E G - H =>Hn Hg is a Cyclic p~group, then H has a normal complement in G if G is solvable or if H is a p-group. The self-normalizing components of a Frobenius cp-partition are called the cp-complements when the conclusion of the above theorem holds. We then prove the following theorem: Let G be a Frobenius cp-partition of a finite group G where one of the self-normalizing components T is a Hall subgroup. Also, assume that G is solvable or T is a p-group. Then, G has a Frobenius cp—kernel K. Moreover, if Z(K) contains a p'-element of composite order, then there is a normal JAMES W. RICHARDS component L of 0‘ such that the following hold: . a) KS L; b) L = K iff O’ is a Frobenius Baer partition; c) If KC Ll, then L/K is a cyclic p-group; d) K is nilpotent unless the Frobenius cp-complements are p-groups. Baer R. I: l] . Partitionen endlicher Gruppen. Math. Z. 75, 333-372 (1961). Copyright by JAMES W. RICHARDS 1969 Cn-PARTITIONS OF FINITE GROUPS BY Jame s W Richards A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1968 ACKNOWLEDGEMENTS The author wishes to express his gratitude to his major professor, Dr. W. E. Deskins, for the immeasurable amount of help furnished by him. Also, he wishes to thank Professor J. Adney for his helpful comments. ii TABLE OF CONTENTS Page INTRODUCTION ......................... 1 CHAPTER 1. Notation and Basic Concepts ............. 6_ II. G-Admissible Subgroups of CP-Partitions ...... 26 III. Frobenius CP-Partitions . . . ............ 41 APPENDIX A . ......................... 54 APPENDIX B .......................... 58 INDEX OF NOTATION . ................... . 62 BIBLIOGRAPHY ....................... . 65 iii INTRODUCTION 1. Background R. Baer [4] defines a partition of a finite group G as a family 0' of subgroups of G such that each ghé 1) EG is contained in exactly one member (or component) of O. The problem of determining all groups which admit non-trivial partitions, that is, with O‘ i5 {G}, has been solved by Baer [1], [Z] [4] , Kegel and Wall [9] , and Kegel [8] The class of all groups which admit non-simple partitions was characterized by Baer [4] . These non-simple partitions are partitions whose groups possess a pr0per normal G-admissible subgroup, i. e., a subgroup K of G satisfying the property that L0 K i I = LE K for all components L of O’. Frobenius partitions are special cases of non-simple partitions. They are partitions which have a proper self-normalizing component. Their structure is described in theorem B. 6 of appendix B. The remaining non-simple partitions are described in theorem B. 3. The other class is the class of all groups admitting simple partitions, i. e. , partitions which possess no proper normal U-admissible subgroup. It turns out that the key to the analysis of this class is the sockel S(G), the product of all minimal normal non-trivial subgroups of G. Baer shows that S(G) must be either abelian or non-abelian simple if G admits a non-trivial Z partition (see theorem B. 7). If S(G) is abelian, thenthe fitting subgroup F(G), the maximal normal nilpotent subgroup of G, is non-trivial. .Baer shows that the only group G which admits a non-trivial simple partition where F(G) 9‘- I is 54 (see theorem B. 2). Some insight into the final determination was given by Baer (see theorem B. 9) when he showed that if S(G) is non-abelian simple, then: [ G:S(G)J = Z, the Sylow p-subgroups for odd primes are abelian, and the Sylow 2-subgroups are D-groups. The problem was completely solved when Kegel and Wall [ 9] showed for S(G) non-abelian that: G is PG L (2, q) where q is an odd prime power > 4 if G is not simple, and G is either one of the Suzuki [l2] groups or PS L(2, q) where q is given as above, if G is simple. 2. Statement of the Problem The purpose of this dissertation is to generalize the concept of partition of a finite group. It should be pointed out that there are many possible directions of generalization. The author's choice is to generalize upon the intersection property. Before we give the formal statement, we need to introduce some notation. If n is a positive integer, then TT [n] denotes the set of all prime divisors of n. An element g of a group G is said to be a TT-element for a given set of primes TT if 3 1T [IgnSTL A subgroup H of G is said to be a n-subgroup of 11 [IHI] ETT. Note that “I: ‘1‘ J_= (5 where I is the trivial sub- group of G and hence, I is a TT-subgroup for any set of primes. 'We now describe the generalized partitions studied in this thesis. If G is a finite group and if TIE TT [‘GIJ , then a family 0‘ of subgroups of G is said to be a cTT-partition of G if (i) O’ is a cover of G, (ii) no member of O' is a proper subgroup of any other element of O’, and (iii) distinct members of O‘ intersect as a cyclic TT-subgroup. The restriction that no member be a proper subgroup of any other is introduced so that redundancy may be eliminated. The general- ization is realized when one sees that a partition as defined by Baer is a c ¢~partition. 3. Objectives and Synopsis of Results The original object of this dissertation was to classify groups which admit non-trivial cTT-partitions. This we have been able to do in certain special cases. In order to obtain a complete classification, two well-known results must be generalized. The first one that must be generalized concerns the Hp-problem, solved by Hughes and Thempson [6] . The needed generalization will be alluded. to in the conclusion of theorem 2. 2.1. The famous Frobenius 4 theorem, see theorem A. 6, is the other result which must be generalized. As shown in Chapter III, no complete generalization can be found, in the sense which we need. However, a partial generalization is obtained. After the definition of ch-partition in chapter I, we discuss the lattice of all possible cTT-partitions of a given group. Next, we consider commuting elements in G and develop sufficiency criteria for elements to be contained in a common unique component of 0'. These conditions enable us to characterize all nilpotent groups which admit non-trivial cTT-partitions when TT is not too large. We finish the chapter with a structure theorem for some groups which admit non-trivial cfi-partitions when suitable restrictions are placed on T1. In chapter II, we first define the concept of O-admissibility in a fashion analogous to that for a Baer partition. We restrict ourselves in this chapter, and throughout the rest of the thesis, to cp-partitions, or cTT-partitions where TT = {p}. After a sequence‘ of lemmas and theorems, we obtain the main result of this chapter which is a structure theorem for all groups admitting a non- trivial cp-partition where all components are normal subgroups. We end the chapter with a theorem which gives some insight into the structure of components which are not self-normalizing sub- groups of G. The last chapter is devoted to Frobenius cp-partitions 5 or those cp-partitions which are normal and contain a proper self-normalizing component. After obtaining a partial generalization of Frobenius's theorem, we use the result to derive information about some Frobenius cp-partitions. A list of symbols and notation used in this thesis is given in the index. Also, there are two appendices. Appendix A is for general group theoretic results while Appendix B is reserved explicitly for results on Baer partitions of groups. CHAPTERI NOTATION AND BASIC CONCEPTS All groups hereunder are assumed to be finite. In this chapter we are concerned with the family of all cTT -partitions of a given group. Also, we consider commutability properties which yield sufficiency criteria for elements to be contained within a unique common component of a given cTT -partition. The latter situation will enable us to characterize all nilpotent groups which admit cTT-partition's when suitable restrictions are placed on TT. Let [n [n] denote the set of all prime divisors of the positive integer n. If 11 is a given set of primes, then a group G is said to be a'fl-group if TTHG‘] E11, and a W-group if HUG‘JQ T1 = ¢. Similarly, an element g of a group G is said to be a TT-element if 1T[ lgllETT, and a T11-element if 11 [‘g‘ Jfln :: ¢ A TT-group (TT-element) is said to be p-group (p-element) if T1 = {p}. We note that the only ¢-group is I, the trivial group. We shall see that a C¢~partition is precisely a partition as defined by R. Baer [4]. Definition 1. 1: Let TIETT [ ‘GH . A family 0' of subgroups of a group G is said to be a CIT—partition of G if the following hold: a) G = U H such that no member of O’ is a proper subgroup H60 of any other b) If H, K” H) E 0', then Hfl K is a cyclic n-subgroup. The members of C will be called components of 0. If G = {G}, then 0 is a CH -partition for any given set of primes. This obviously trivial cTT-partition will be referred to as the trivial CTT-partition of G. If TT= {p}, then 0 will be called a cp-partition. R. Baer [4] defines a partition of G as a family 0‘ of ‘ subgroups such that g(# 1) E G implies that g is contained in exactly one member of 0‘. Then, distinct members of 0’ intersect as the trivial subgroup I, the cyclic Qi-subgroup of G. Also, no member of O’ is a preper subgroup of any other. So, 0' is a c¢-partition. Conyersely, it is easy to see that any c¢-partition of G is a partition as defined by Baer. We hence- forth shall refer to these c¢-partitions as Baer partitions. Let G be any non-cyclic group, T the family of all maximal cyclic subgroups of G, and TT* fl Cl]: U {nUHflKlj : H, K(#H) ET}. Then, T is a CTT-partition where 11: Wk [‘G‘] . Hence, if TT* “CHEN ETT [ lGl] and if G is non-cyclic, then G always admits a CTT-partition, namely, the one above. Furthermore, it is clear that T is non-trivial since G is not cyclic. If G is cyclic, then the subgroups of G are well-ordered and thus, it is impossible for G to admit a non—trivial CTT-partition for any 8 choice of 11. Consequently, in order to restrict our attention to a class of groups, the members of which do not necessarily admit a cTT-partition, we shall usually assume that 11* “CUE 1T. In particular, we will have 11 = {p} in the later sections. Example 1.1.1: Let G ‘= A 5 and 0‘ be the family of all maximal cyclic subgroups. Since A5 contains no elements of composite order, we have that 11* [)AS‘] = 05. So, 0 is a Baer partition of A5. Example 1.1. 2: Let G = A7. The elements of A7 are as follows: Type Order (1,2,3,4,5,6, 7) (1,2,3,4,5) (l,2,3,4) (5,6) (1,2,3) (4, 5) (6,7) (1,2,3) (4,5,6) (1,2,3) (1.2) (3.4) (1) HNWWO‘QUTQ Now, let T be the family of all maximal cyclic subgroups of A7. It is evident that TPv" HA?) J: [2}. So, T is a CZ-partition of A7. N. Iwahori and T. Kondo-- [7] , theorem 5--have shown that the alternating group An on n elements admits a Baer partition if and only if n = 4, 5, or 6. So, A admits a cZ-partition, but 7 not a Baer partition. 9 We shall now obtain a means of comparing two cfi-partitions of a group G. In particular, we shall introduce a partial ordering on the family of. all cTT -partitions of G, where 11 ranges over all subsets of 11[|GI] . Definition 1. 2. Let 01 and 0‘2 be 03- and c112 -partitions respectively of a group G. Then, Ci is said to be a refinement of 0‘2 if each component of 01 is a subgroup of some component . ' < e ' e ' ' < of 02 We write 01 _ 2 1f 1 is a refinement of U2 and 0‘1 02 for preper refinement where 01502 but 01 1‘ 02. It is clear that the concept of refinement defines a partial ordering on the family of all cfi -partitions of G since inclusion is a partial ordering on the family of all subgroups of G. If G is not-cyclic, then denote by pm(G) the CU -partition of G which consist of all.the maximal cyclic subgroups of G and by pM(G), the trivial cTT-partition of G. (When there is no danger of ambiguity, we simply write pm and OM respectively.) Similarly, write rs: for 115i [101] . Now, if a is any cTT-partition of G, then we see that £3me: (3M. So, the family of all CWT-partitions of a non~cyclic group always has a largest element and a smallest element. We now proceed to establish that the family of all cfi—partitions of G is indeed a lattice under our partial ordering. In fact, we 10 shall describe the meet and join operations in a precise fashion. Theorem I. 2.1: Let 01 and 02 be cnl-and c112 -partitions respectively of G. Then, the maximal elements of S = (HflK:HE01, K E02} determine a cfi3-partition, written 3 " U = . ' 01A 0‘3 of G where 1 TT2 113 and 0'1 A 0'3 15 the largest refinement of both 01 and 02. Proof: Let 0‘ A02 denote the family of all maximal elements of l S = {HOK:H Eel, K E02}. It is clear that the numbers of 01 A02 are subgroups of G. Let g EG. Since 01 and 02 are both cTT-partitions of G, there are subgroups H and K of G such that gEHEO and gEKEo Then, gEHflKES implies thatthere 1 2° is a maximal element H10 K1 of S such that g E H10 K But, 1. HflK E0 A0 and hence, G =U L. Since 0 A0 consists l l l 2 , LEO A0. 1 2 1 Z precisely of the maximal elements of S, no member can be a proper subgroup of any other. Now, let U, V(;E U) 601 A0 Then, U = HIGH and 2' 2 = n V K1 K2 where H1, K1 E31 and H2, K2 E02. But, U )5 V implies that Hl aé Kl or HZ if K If Hl # K1, then U n v = 2. . ' n C 0 ° - lo . (H10 K2) fl (K1 K2)__ H1 Kl a cyclic 1'3 su agroup The other part follows mutatis mutandis. Thus, the intersection of any two distinct members is a cyclic n3—subgroup where 113: 111 UTTZ. Therefore, 0'1 A 02 is a cTT3-partition of G. 11 Now, let T be any cTT-partition which is a refinement of both 01 and 52. Let L ET. Then, T501 and T: 02 imply that there are subgroups H of G such that LSH EO and l 2 l 1 LE Hz E02. $0, LSHl ’1 H2 E S. It then follows that there is a and H maximal element U of S such that LE U. But, UEolAOZ. So, LEU E01 A0 , which implies that Tie Ao . Therefore, 2 1 2 01 A02 is the largest refinement of both 0‘1 and 02 and the proof is completed. The refinement 0‘1 A 02 of both 01 and C32 will be referred to as the meet of Cl and 02. Next, .we wish to define the join of two CWT-partitions. If 0‘1 and 02 are two given chl- and cTT‘2 -partitions respectively of G, then they both are refinements of p M' So, the obvious approach is to define the join of 01 and 02 as the meet of all cTT-partitions, each one of which 01 and 02 are refinements. This approach is contingent on the fact that the meet Operation is associative and hence, our next theorem. Corollary 1. 2. 2: If oi is a cTTi-partition for i = 1,2, 3, then (01 A02) A o 3 = 01 A ((32 A03). Proof: Let W E (01 A02) A03. It then follows by theorem 1. 2.1 that there are subgroups Hi of G such that W = (H10 H2) 0 H3 where H,Eo, for i=1,2,3 and H UH EU AO‘ . Since 0' AG consists 1 1 Z 1 2 Z 3 l 12 of the maximal elements of S = {K flK : K EO‘ , K E 0‘3}, we 2 3 Z 2 3 have that there is a U E0 A 03 such that W = Hlfl (Hzfl H3) E Hzfl H 2 3 S U. It now follows that (01A 02) AG :0 A0 . We also have 3 2 3 that W = (Hl fl H2) fl H E H EO’ , which implies that 3 l 1 < < (0‘1 A02) A0 _01. We then conclude that (01 A02)A 03 ~01 A (02 A 03) 3 o A O o smce 01 A (02 0'3) 13 the largest refinement of 01 and 0‘2 A03, and (0‘1 A02) A03 18 a refinement of 01 and 02 A03. By repeating the same type of argument as above, we see that 0‘1 A (OZAO3) : (01A 0‘2)ch3 . The proof is then complete. Since we now know that the meet Operation is associative, we may consider without difficulty a refinement of any non-empty Collection (Ci: 1: if n] of cfii-partitions of G. Thus, we have our next theorem, the proof of which we omit since it is straight forward by induction. Corollary 1. Z. 3: Let (oi: If if n} be any non-empty collection of cn,-partitions of G. Then, T = A 0', is a CU -partition of G 1 . 1 s 15 l_<_ n n where ns = 191 Hi and the components of T are the maximal n elements of S = [ fl Hi: Hi EO‘i}. Moreover, T is the largest i=1 refinement of the oi's. We now define the join operation. 13 Definition 1. 2.1: Let T1 and T2 be cTTl- and CITZ -partitions of G and 2 be the family of all cn-partitions of G. Then, T : < ° : IVTZ Moiloiez, Tj— oi, J 1, 2}. Our next theorem establishes the fact that Z is a lattice under our partial ordering. We omit its proof. Theorem 1. 2. 1: If T1 and T2 are cTTl- and cTTZ-partitions of G respectively, then Tl VT2 is the smallest CIT-partition of which T1 and T2 are both refinements. Let 0‘ be a cTT-partition of a group G and B be a group of automorphisms of G. Let B E B and H E0. Although H8 is a subgroup of G, H8 is not necessarily a component of 0‘. This suggests the following definition. Definition 1. 3: Let 0 be a cTT-partition of G and B be a group of automorphisms of G. Then, G is said to be B-invariant if HBEO‘ whenever H EO’ and 8 E B. O' is said to be normal if B = I(G). Theorem 1. 3.1: Let G be a CTT—partition of G and B a group of automorphisms of G. Then, 0 has a B-invariant refinement which is a CIT-partition. ~ '3 Proc_>_f: For B E B define 0'8: {H’ :H ECT}. Now, 0'8 is clearly a ct'c-partition of G for each 5 EB. Define T = A 08. We have 14 by corollary l. 2. 3 that T consists precisely of the maximal elements of S ={ fl H :H E08}. Now, let HET. Then, a e B 5 B H: fl K where K E08. But,KEO‘ B B: K” 60511 for each BEB B B p. E B. 50, K: = Lp E Up for some pE B. Hence H“L = n K” = n L which implies that H” E 5. Since BEB B p63 7 automorphisms preserve the lattice structure of G, we have that H” must be a maximal element of S. Thus, H“ E T, which establishes that IT is B-invariant. The fact that 7' is a CT? -partition follows directly from corollary l. 2. 3 since 118 = n for all 8 E B. The proof is now complete. The techniques used to prove the preceeding theorems are analogous to the ones used by Baer and credited by him to O. Kegel--see Baer [4] , pages 359-351. We shall now develop some sufficiency criteria for elements of a group to be contained within a common unique component of a CIT-partition of G. The following theorems in this connection are analogous to Baer's criterion for Baer partitions and in fact the same type of argument is employed for each one--see theorem B. 1. Let g be an element of a group G and TTETT [|G|J. Then, 15 lg( factors uniquely as nin where U [m] ETT, 1T[n] flfi = ¢; consequently, (m, n) = 1. It then follows by theorem A.l that g = glg.2 = gzg1 uniquely where (gl| = m and gal = n. Also, g1 and g2 are both powers of g. gl is said to be the TT-part of g and g2 the Tit-part. This decomposition of g is called the TT-decomposition of g. Theorem 1. 4.1: Let G be a cTT-partition of a group G and x,y be elements of G which commute. Then, there is a unique component of e which contains both x and y if one of their Tfl-parts has composite) order. M: Let XlXZ = x and yly2 = y be the decompositions of x and y into their respective TT-parts and TT'-parts. Let us first consider the case where ‘yzl < lle. Now, xy = yx implies that xiyj = iji for i, j = l, 2 since powers of commuting elements commute. Let m = (xlyl . 0 being a CTT-partition implies that there are components H and K of e such that xE H and xy EK. But, .(lxlyll, ‘xzyzh =1 and lyzl < (x2) together imply that 1 *- xzm = (xy)m E H fl K. Then, we conclude that H = K since xzm is a TT'-element. So, x, xyEH. But, H . -1 being a subgroup of G yields that y = x (xy) E H. The uniqueness of H follows from the fact that x E H and x is not a TT~element. We get the same conclusion when lle < (yzl 16 mutati s mutandi s . Let us assume now that Ile = (yzl. Since x or yZ has 2 composite order and (x2) = lyzl , we have that x2 and y2 both have composite order. Let q be a prime divisor of Iyzl and z = ylyzq. Then, z = ylz2 where 1 15 z q. Also, lzzl< (x2). 2 = y2 It then follows from the previous paragraph that there is a unique component H of 0‘ such that x, 2 EH. Also, y2 being of composite order and ‘22) < lyzl imply by the previous paragraph that there is a unique component K of G such that z, y E K. Then, z EH flK and hence, H = K since z is not a n-element. The proof is now complete. The assumption that x or y2 has composite order in the 2 preceding theorem cannot be removed as the following example shows. Example 1. 4. 1: Let G be the abelian group of order 18 whose Sylow 3-subgroup is elementary abelian. The family of all cyclic subgroups of order 6 determines a cZ-partition of G. Let and be distinct components of 0.. Now, xy = yx, but x and y are contained within distinct components. The 2'-parts of x and y both have order 3. So, theorem 1. 4.1 cannot be improved. The above theorem gives us a sufficiency criterion, in terms of the fi'-parts, for elements to be contained in a unique common 17 component. Our next step is to develop one in terms of the TT-parts. Recall that the exponent of a group G, written exp(G), is the smallest positive integer such that gn= l for all g E Gm Definition 1.4: Let 0' be a CTI-partition of a group G. The exponent of o is defined to be exp(G) = l. c. m. {exp(HflK): H, K(ié H) E0} ifO‘ is non-trivial and exp(G) = exp(G) otherwise. If 0' is non-trivial, then it is clear that 0 is a Baer partition if and only if exp(O‘) = 1. So, exp(G) in some sense measures how close 0 is to being a Baer partition. Theorem 1. 4. 2: Let U be a cfi-partition of G and x, y be commuting elements of G. If lxl‘ > Iyll exp(O‘) or (y1l>(xllexp(0‘) where x1 and y1 are the TT-parts of x and y respectively, then there is a unique component H of O‘ which contains both x and y. Proof: Let us assume that (x11 > (yll exp(G) where x = xlxz and y = yly2 are the decompositions of x and y respectively into [their TT-parts and TT'-parts. Let H and K be components of e .such that x EH and xy EK. Also, let m = lxzyz‘ and k = (yl‘. Now, (m, ‘th = 1 implies that (xlml = (x1). This together with k: k exp(G) < \xll yields that 1 if xink= xmk m)k (“Mon 1 Since : (Xy)mk 6 H (1 K. However. (X1 18 m l Ix I = (XI! > k exp(O‘). Hence, exp(H fl K) does not divide exp(G) which implies that H = K. So, x, xy E H which implies that x, y E H since H is a subgroup. The argument for the case lyll >lx1‘ exp(G) is the same as before. Assume that there is a component L of e which contains both x and y’ and such that L15 H. Then, EHfl L, a cyclic TT-Subgroup. In particular, it follows that x = x1 and y = y1. Let w be a generator of . If lxll> )yll exp(G), then (w): lle > exp(G) which is a contradiction to the fact that exp (H fl K) divides exp(G). We get the same contradiction if (yll > (xllexpw). So, we are forced to conclude that H is indeed unique and hence, the proof is complete. The assumption that |x1| > lyll exp(G) ‘ or “’2‘ > le) exp(G) cannot be weakened as the following example shows. Example 1.4.2: Let G = H x Q where lH| = 2 and Q is the quaternion group of order 8. exp(G) = 4. Let G be the family of all subgroups of order 4. Then, 0' is a cZ-partition of G and exp(G) = 2. Let x E Q have order 4 and y E H have order 2. Then, xy = yx. Now, x and y are contained in distinct components of 0. However, ‘xl = ly]exp(0). This shows that the theorem cannot be improved. We now examine some consequences of the above theorems. 19 Theorem 1.4. 3: Let U be a cTT-partition of a group G and H, be a subgroup of G. H is then contained in a unique component of 0 if Z(H) contains a TT'-element of composite order. M: Let h E Z (H) be a Tf-element of composite order and K be the unique component of O which contains h. Let x E H. Since h E Z(H), we have that xh = hx. Since h has composite order, we conclude from theorem 1. 4.1 that x and h are contained within a unique component of 0‘. But, K is the unique component of O which contains h. So, x E K. Hence, H E K. The uniqueness of K follows from the fact that it is the only component which contains h. The proof is now complete. Corollary 1.4.4: Let G be a group and nEn[ |C|]. If Z(G) contains a TT'-element of composite order, then G admits'only the trivial clT-partition. Proof: Let 0’ be any cTT-partition of G and g0 be aW-element of G_ of composite order such that goE Z(G). It then follows by theorem 1. 4. 3 that G is contained in a unique component of 0. But, this occurs when and only when 0' = {G}. So, the proof is complete. Our next step is to extend these theorems to a slightly more general Situation. Let G be a group and 20 = I. We inductively define Zi by 20 zit/21”1 = z(G/zi_1). Then, I 2 z0 :21: . . .Zn= zn+1= Zn+2= since G is finite. This series is called the upper central series of G and its largest member is called the hypercenter h(G) of G. Another characterization of h(G) given by Baer--see theorem A. 2-- is that an element g of G is contained in the hypercenter if and only if each p-part of g is centralized by all p'-elements of G for every prime p. Also, ’we mention here that G is nilpotent if and only if G = h(G). Now, let 0 be a cTT-partition of G and U be a subgroup of G. If S = {K fl U:K EO‘}, then it is clear that the maXimal elements of S determine a c110 -partition of U where HOE“. This c110- _ partition is called the induced partition on U by O' and‘is denoted as O' . Obviously, if U is not contained in a component of 0', U then 0 and GU both are non-trivial. We now describe the structure of h(G) if it is not contained in a component of 0. Theorem 1. 4. 5: Let G be a cTT-partition of a group G and assume that h(G) is not a TT-subgroup and is not contained in a component of 0'. Then, h(G) is the direct product of a cyclic lT-subgroup with a group whose center has exponent q. Proof: Since h(G) is nilpotent and is not a TT—subgroup, we know that (h(G) = H x K where H is a TT-Subgroup and K is a TT'-Subgroup. Clearly, Oinduces a non-trivial cTT -partition 0 0 U 21 on U = h(G). Now, Z(U) is not a fi-subgroup since n[ [Uh = n[12(U)|] . It then follows by corollary 1.4.4 that exp(Z(K)) = q, a prime. We assert now that if k(ié l) E K and if k E L ECU,then HE L. Let h E H. The conditions: U nilpotent, k a TT'--element, and h a TT-element, imply that hk = kh and that hk is contained within. aunique component T of O From ((hl, (kl) =1 and U° lh‘ = n, we conclude that l :9 kn = (hk)n E Lfl T. It then follows that L = T since kn is not a fi-element. So, k, hk E L and hence, h = (hk)k“l E L since L is a subgroup of G. Thus, the assertion that HS L is established. Now,O’Uis non-trivial and so there is a component M of 0U such that M i5 L. Since U = H x K where H is a TT-subgroup and exp(z(K)) = q, we have that M = Hl x Kl where HIE H and K1: K. It then follows from the fact that H g L if M that M contains an element m of order q such that m EM - L. Since (ml = q implies that m E K, we see that HE M. So, HEM n L such that M :9 L implies that H is a cyclic n-subgroup. Therefore, the proof is complete. In the proof of the next theorem, we use some results on Baer partitions. The needed concepts are the following: Definition: If 0‘ is a Baer partition, then a subgroup K of G is said to be "CT-admissible if Lfl K if I =9 LEK for all components 22 L of 0. Definition: A Baer partition of of G is said to be non-simple if G contains a preper normal o-admissible subgroup; otherwise, it is said to be simple. Definition: A Baer partition 0 is said to be Frobenius if 0 contains a self-normalizing component which is proper. We now state the main theorem of this chapter. Theorem 1. 4. 6: Let G be a nilpotent group and TTCTT [lG‘J . Then G admits a non-trivial cTT-partition if and only if G is the direct product of a cyclic TT-group with a Sylow q-subgroup Q of G such that quQ) C Q and IQI :6 q. _P_r_op_f_: Let us begin by assuming that G admits a non-trivial CIT-partition o. By theorem 1. 3.1, we may assume that 0‘ is normal. Since G = h(G), it follows by theorem 1. 4. 5 that 11 [IG‘J = TTU {q} where qIETT, and that G = H x Q where H is a cyclic TT-subgroup and Q is the Sylow q-subgroup of G. We assert that Q is not contained in any component of 0‘. Assume that there is a component L of G such that Of L. Let x(7é 1) E Q. As in the proof of theorem 1.4. 5, we see that HE L. However, this implies that G = H x Q = EL, a contradiction to 0‘ being non-trivial. Thus, Q is contained in 23 no component of 0. It now follows that O' induces a non-trivial cTT-partition O on Q. However, Q contains no TT-element of G other Q than 1. So, 0Q is in fact a Baer partition of Q. Also, 0‘ Q is normal since 0Q is the maximal elements of S = {Qfl L:L EU} and O is normal. From the fact that Q = F(Q), we conclude by theorem B. 2 that o is not simple. Also, Q 00 is not Frobenius since nilpotent groups contain no proper self-normalizing subgroups. So, Q contains a proper normal 0Q -admissible subgroup K. If g E Q - K, then it follows by theorem B. 3(b) that Ig‘ = q. This implies that ‘quQ) E KCQ. It is clear that lQl if q since 0 is non-trivial on Q. This Q completes the only if part. Let us assume now that G = H x Q where H is a cyclic n-group and Hq(Q) co where [Q] a! q. By theorem 13.4, it follows that Q admits a non-trivial Baer partition 0. Define r = {HLz LEO}. Now, if K and L are subgroupsof Q, then HLE HK if and only if LE K. Let g E G and g = g be the decomposition of g into its lgz respective TY-part and TT'--part. Since G is a Baer partition of Q, there is a component L of 0‘ such that g2 E L. Thus, = gng E HL ET. If HL ,L HK, then it is clear that H: HLleK, This completes the if part and hence, the theorem. 24 We shall now describe the nature of a cTT-partition 0‘ of G when suitable restrictions are placed on the given set of primes. Theorem 1. 4. 7: Let 0' be a non-trivial cTT-partition of G such that nfl TT [[ G:h(G) J] = ¢. Then, there is a normal Hall TT-subgroup H of G such that T = [L/H: L E0} is a Baer partition of G/H. M: Let us first consider the case where G is nilpotent. Then, G = h(G). If G has no fi-elements, then 0 is a Baer partition of G. So, assume that G contains TT-elements and let H be the normal Hall 11 -Subgroup of G. We showed in theorem 1. 4. 6 that H must be contained in all components of 0. Let L,K(ié L) Eo. Then, H SLfl K. However, H being the maximal TT-subgroup of G and L fl K being a cyclic TT-subgroup imply that H = LflK. So, '1‘: Lfl K/H = L/HnK/H. Let Y x H E G/H. Now, 0' being a cTT-partition of G implies that there is a component S of 0' such that x ES. So, x H ES/H since HE S. Therefore, T = {L/HzL E0} is a Baer partition of G/H. Assume now that G is non-nilpotent. Then, h(G)CG. Let H be the Hall TT—subgroup of h(G). Then, it [[G:h(G)]]’l n = (6 implies that H is the Hall TT~subgroup of G. Also, G contains a Tf-element x3é 1. Let L be the component of 0' which contains x. Let h E HEh(G). It then follows by theorem A. 2 that xh = hx. If K is the component of O’ which contains hx, 25 we then have for lh‘ = n that = = <(xy)n> S Lfl K since x and h are coprime elements. But xfiE 1) being a Tf-element implies that L = K. So, x, xh E L which yields that h = x-l(xh) E L. So, HE L. Let L and K be distinct components of 0. Using the same argument as in the first - paragraph, we have that T = L fl K/H = L/H fl K/H and hence, T = {L/H: L E0} is a Baer partition of G/H. This completes the proof. If the restriction placed upon the given set of primes is removed, then the theorem is false as the following example shows . Example 1. 4. 3: Let H = . Let 6 be the automorphism of H defined by: a9 = a, be = b-l, c6 = c‘l. Let G be the extension of H by 9 . Since ‘9‘ = 2, we have that H = h(G). We define a CZ-partition 0‘ of G to be the family of all Sylow 2-subgroups tOgether with all the cyclic groups of order 6. We see that {2} = fl = fi[[G:h(G) ]] . Since G does not have a normal Sylow Z-subgroup, the conclusion of theorem 1. 4. 7 is not satisfied. CHAPTER II 0‘ ~ADMISSIBLE SUBGROUPS OF CP-PARTITIONS In this chapter, and throughout the rest of the thesis, we consider cp-partitions where p is a fixed but arbitrary prime. Definition 2.1: Let 0 be a cTT-partition of a group G. Then, a subgroup H of G is said to be G-admissible if for each component L of 0 either Lfl H is a cyclic TT-subgroup with exp (Lle) dividing exp (0) or LE H. I All non-cyclic components of a cTT-partition are U-admissible subgroups but the converse is not true as the following example shows. Example 2.1.1: Let A = where ‘al = 4 and G = A x S 3. The family 6 of subgroups of order 4 of G together with the subgroups of order 3 determine a c2-partition of G where exp (0‘) = 2. The Sylow Z-subgroups of G are U-admissible but are not components of G. If G is a normal cTT-partition of G and if H is arr-admissible subgroup of G, then any conjugate of H is O-admissible. However, H if Hg does not imply that H fl Hg is a cyclic W—subgroup as the Z6 27 following example Show 3 . Example 2. 1. 2: Let G = S4 and define C to be all the conjugates of S3 in S4 together with all the cyclic subgroups of orders equal to 4. 0' is a (:2 -partition of G and exp (0) = 2. The Sylow 2-subgroups of G are G-admissible but intersect in the Klein 4-group which is not cyclic. We now begin with a sequence of lemmas and theorems which will terminate with a structure theorem for those groups which admit non—trivial cp-partitions having all components normal. Theorem 2.1. 1: Let C be a cp-partition of a group G, K a o-admissible subgroup of G, U a component of O, and u E U - K. Let u = uluz be the decomposition of 11 into its respective p-part and p'--part. Then, the following hold: a) If uz =1, then exp (CK(u)) divides lull exp (o); b) If lull = q, a prime, then C (u) is the semi-direct K product of a subgroup of U fl K by a group of exponent qor 1‘; c) If lug! is composite, then CK(u)C_:UflK. Proof: Let us first consider the casewhere u2 = 1. Then, u = ul =uEU-K. Let kECK(u)andk=kk which implies that u l 2 1 be the decomposition of k into its respective p-part and p'-part. 28 Since k1 and k2 are both powers of k, it follows that k1,k2 E K. H k2 if 1, then there is a unique component S of e which contains k2. But, Kbeing o-admissible and szKflS imply that SEK since kZ 9‘ l is a p'-element. Since k centralizes u, we have that k2 centralizes 11 because k2 is a power of k. Also, k2 and u are c0prime elements of G. Let n = lul and W be the unique component of U which contains ukz. Now, = = <( usz‘ > E w r) K. Then, K being o-admissible and k2 being p'-element imply that WE K. However, if -m = lkzl, then < u> = = < (uk2)m> S K, a contradiction to our assumption that u E U - K. Hence, k2 = l. ' Assume now that lkl = lkll > lullexp (0). It then follows by theorem 1. 4. 2 that there is a unique component R of O‘ which contains both 111 and k1. But, kl E R fl K and exp (0) < lull exp (0) < lk imply that exp (R fl K) does not divide exp (0‘) 1l and hence, RE K. Then, it follows that u = u E RE K, a l contradiction to our assumption that u E U - K. Therefore, we see that exp (CK(u)) does indeed divide lull exp ('3) which completes part a. Assume that luzl = q, a prime. It is clear that CK(u)C_'—‘_ CK(u2). Let kECK(u2). If k is a p-element, then let W be the component of U which contains ku = u k. Let m = . lkl . 2 2 Then, we have = = <(ku2) m> E w n U. But, w n U is not a cyclic p—group and so W = U. Since k Ef§ U, we conclude . 29 that . k E U fl Iii. It then follows that CK(u2) has a unique cyclic Sylow p-Subgroup contained in K fl U. Now, let k be any p'-element 01f CK(u2). Since U fl K is a cyclic p-group, we know that 1:1: U. So, it follows by theorem 1.4.1 that lkl = r, r a prime. If r 4 q, then, ku has composite order and 2 centralizes. k. 1 But, this implies by theorem 1. 4.1 that kuz E- U and hence, k = (licuZ)uZ-l E U, a contradiction. So, lkl = q. Thus, we have that CK(u) is the semi-direct product of a subgroup of U fl K by a group: of exponent q. Finally, assume that luzl is composite. It then follows by theorem 1. 4.1 that. CK(uZ) E U and hence, CK(uzlS K fl U. However, C .; ' ' ' , C fl CK(u)_ CICuZ) Since u is a power of 11 Hence, CK(u) __ K U and 2 the proof of the theorem is complete. Corollary 2.1. 2: Let G be a cp-partition of a group G, .K a O-admissib-le subgroup of G, and U a component of 0‘. Let u E U - K such that u is neither a p-element nor a p'-e1ement. Then, CK(u-)g Ufl K. Proof: Let. u = uluZ be the decomposition of u into its respective p-part and p“-part. Then, u being neither a p-element nor a p”element implies that 111 f 1 ii 11 If luzl is not a prime, then 2' we have» by theorem 1. 2.1 (C) that CK(u)E U fl K. So, assume that luzl = q, C}? a prime. It then follows by theorem 2. 1.1 (b) that CK(u) is the semi-direct product of a subgroup of U fl K by a 30 group of exponent q or 1. But, we have by theoremZ. 1.1 (a) that exp (CK(ul) ) divides l ull exp (0) which is a power of p. Since CK(u)E CK(ul)’ we now conclude that CK(u)S U fl K which completes the proof. The next two lemmas which we prove are well known facts about automorphisms. Lemma 2.1.3: Let ‘y be an automorphism of order n> l of a group G and assume that H if I is a normal 'y-invariant subgroup such that - YH = idH and y induces the identity automorphism on G/H. Then. there is an; lab19 1) E H such that hn= 1. Proof: Let {g1 = l, g . . , gt} be a complete set of coset 2" representatives for H in G. Since n >1, there is a gi such that 7(gi) :5 gi. Since 7 induces the identity automorphism on G/H, it follows 7(gi) = gih where h (f 1)E H. Since y(h) = h, gi = 7n(gi) = gi hn which implies that hn= l and hence, the proof is complete. Lemma 2.1. 4: Let y be an automorphism of order 11 >1 of G and G)’ = (g E G: ‘y(g) = g} be the stability group of 1G relative to y. If G7 is. normal and if n is coprime to lel , then ‘y induces a fixed-point-free automorphism on- G/G . Proof: Since 7 35 idG, it follows that G)! C G. Assume that 31 ‘y(g)Gy = gGy for some g E G - Gy and define H = . Clearly, H is a ‘y-invariant subgroup of G and 7H :5 idH since GYC H. But, 7H induces the identity automorphism on H/Gy' So, we have by lemma 2.1.3 that there is a gghé 1) 6 Cy such that gom= 1 where m is the order of )IH. However, m divides n which is a contradiction to the assumption that n and lel are coprime. Thus, ‘y induces a fixed-point-free automorphism on G/H and the proof is complete. Lemma 2. l. 5: Let G be a cp-partition of a group G, U a compo- nent of U, and K a o~admissible subgroup of G with UZ K. Let T = KflU. If x(# l) EUhas order qfip and if.M is an x-invariant subgroup of K which contains T as a preper normal subgroup such that x induces the identity automorphism on M/T, then q divides [MzTJ . _I-_’r_og_f_: Assume that q does not divide [M:T] as described in the hypotheses. Let C = CK(x) where x E U and lxl = q, a prime 3‘ p. We have by theorem 2.1.1 (b) that C is the. semi-direct product of a subgroup of T = U fl K by a group of exponent q or 1. Hence, CM = C fl M is the semi-direct product of a subgroup of T by a group of exponent .q or 1. Since T is a cyclic p-group and q(# p) does not divide [MzT] , q does not divide lMl .' Hence, CME T. Since T is a cyclic p-group normal in M and CME T, it follows that CM is normal in M Since it is characteristic in T. 32 Now, lxl coprime to lCMl implies by lemma 2.1. 4 that x induces a fixed-point-free automorphism if on M/C . Since T M is a normal x-invariant subgroup of M, it follows that T/CM is a normal S'E-invariant subgroup of M/CM. So, 32' induces a fixed- point-free automorphism (,0 on M/CM/T/CM by theorem A.4 _ x where Cp(mCMT/CM) - x(mCM)T/CM - m CmT/CM. Now. by theorem A. 5, the mapping sz/T rM/CM/T/CM defined by f(mT) = mC T/CM is an operator isomorphism of M M/T onto ki/CM/T/CM. Let m E M - T. Since x induces the identity automorphism on M/T, it follows that mxT = mT. But, f(mxT) = cpffmT) = Cp(mCM)/TfCM )5 mCM/T/CM = f(mT) since cp is a fixed-point-free automorphism of M/CM/T/CM. We then have that f(mT) if f(mxT) = f(mT) a contradiction. Hence, we are forced to conclude that indeed q divides [MzT] and the' proof is complete. Theorem 2.1. 6: Let 0‘ be a cp-partition of a group G, K a o—admissible subgroup of G, and U a component of 0’ such that UE K, but UCNG(K). Then, the following hold: a) [U, NK(U)]EKfl U b) T = K fl U is a normal cyclic: p-subgroup of both U ‘and N = NK(U) c) U/T and N/T are either both p-groups or q—groups of exponent qip if TCN. 33 Proof: Let u E U and k E N = NK(U). Then, uk E U which implies that [u, k] = u-luk E U. Since UC NG(K), we have. lk-luk E K. By that u-lk-luEK and hence, [u, k] = 1,1- combining our results, we have that [U, N] EU fl K which completes part a. Since K is a G-admissible subgroup of G and does not contain U, it follows that T = U fl K is a cyclic p-group. It follows directly from UCNG(K) that T = U fl K is a normal subgroup of U. Consider t E T and k E N. By part a it follows that [t, k] ET. Thus, t [t, k] = k-lt kET. Thus, T is also normal in N. This establishes part b. Let us assume now that ‘TCN = NK(U). We shall show that N/T and U/T are both p-groups if p divides [N:T] and then Show that they are both q-groups of exponent q i5 p if p Idoes _ not divide [N:T] . We first consider the case where p divides [N:T] . We assert that U/T is a p-group. Assume to the contrary and let u E U have order q 1‘ p. Let Sp be a Sylow p-subgroup of N. Then, p dividing [N:T] implies that T CSP. Since [U,N]S T, it follows that SP is a u-invariant subgroup of N and also that S contains T as a normal u-invariant subgroup. But, [N, TJET implies that u induces the identity automorphism on Sp/T. We then have by lemma 2.1. 5, since u {f K, that q(9E p) divides [Sp:T] , a contradiction. So, U/T is a p—group as asserted. 34 We assert next that N/T is a p-group. Assume the contrary and let n E N have order q if! p. Let W be the component of 0 which contains 11. Then, n E N fl WE K fl W which implies that WSK since K is o-admissible and n is a p'-element. So, n E U. Since n E N = NK(U), it follows that U is a n-invariant subgroup of G. Now, T is a normal n-invariant subgroup of U since it is normal in both N and U. We conclude from [U,N]_C_ T that n induces the identity automorphism on U/T and hence, it follows from lemma 2.1. 5 that q divides [UzTJ = pa, a contra- diction to q if p.‘ Hence, it follows that U/T is a p-group. Assume now that p does not divide [N:T].. Let n E N have order q, a prime, and W be a component of C which contains n. Then, 11 E N fl WEK fl W which implies that WE K. since K is G-admissible and n is a p'-element. Now, W fl UEK fl U = T a cyclic p-group yields that n E U. Let H be any subgroup of . U such that TCH. Since [N, U] cT, we have that H is a 4 n-invariant subgroup of U and that n induces the identity automorphisms on U/T and hence on H/T. From lemma 2.1. 5 we conclude that q divides [HzT] . Since H is arbitrary, it- follows that U/T is a q-group. Now, let u E U have order q. Since U fl K = T a cyclic p-group, we see that u E K- Let M be any subgroup of N such that T CM. Now, M is a u-invariant subgroup of N since [N, U] ST. Also, 11 induces the identity automorphism on 35 M/T since [N, UJET. Thus, it follows by lemma 2.1.5. that q divides [MzT] . Therefore, N is a q-group since M is arbitrary. We assert now that exp (N/T) = exp (U/T) = q. Let n E N such that lnl = qz. It then follows by theorem 2.1.! (c) that C = Cu(n)S T. Since T is cyclic and is normal in U we have that C is characteristic in T and hence, normal in U. But, lnl is coprime to lCl. So, we have by lemma 2.1. 4 that n induces a fixed-point-free automorphism of q power order on U/T. However, .it is well-known that a fixed-point-free automorphism (of prime power order has coprime order to that-of the group, which is a contradiction. So, exp (N/T) = q. Now, let 11 EU have order qz. Again, we have by theorem 2. 1.1 (c) that C = CKfu)EKfl U = T. 50, u induces a fixed point free automorphism on N/C. But this is a contradiction since q divides [N:C] and the fixed—point-free automorphism is of q power order. Thus, exp (U/T) = q and the proof is complete. We now prove the main result of this chapter. Corollary 2.1‘. 7: Let 0 be a non-trivial cp-partition of G such that all components are normal subgroups of G. Then, G is either a p-group or the semi~direct product of a cyclic p-group by an elementary abelian q-group. 36 1_3_1_'_c_>_<_)_f_: If G is a p-group, then there is nothing to prove. So, assume that G is not a p-group and let g E G have order q f p. Let L be the component of 0 which contains g. Since 0' is non-trivial, there is a component W of 0‘ such that W ’5 L. So, W fl L = ~T is a proper subgroup of both W and L. Since W and L are normal subgroups of C, it follows that w n L: L = NL(T) and w n 'Lcw = NW(T). Since q divides lLl, we conclude from theorem 2.1.6 (c) that W/T and L/T are both groups of exponent q. We now separate the argument into two cases; (i) W fl L = I and (ii) WflLi‘I. If WflL=I, then W and L are both, q-groups of exponent q. Let R be any component of 0' such that R f L. Then, R fl L being a cyclic p-group and L having exponent q yield that R fl L = I and hence R has exponent q ”by theorem 2.1. 6 (c). So, all components of 0‘ have exponent q. This implies that 0’ is a Baer partition of G since distinct components intersect as cyclic p-groups and G has no non- trivial p-elements. It then follows by theorem B. 5 that G is an elementary abelian q-group. Assume now that W fl L = T f I. Let R1 be any component of 0‘ where R f L. By theorem 2.1. 6 (c) R/RflL and L/RflL are both groups of exponent q. But, this implies that TSR fl L. Then, we conclude that T = R fl L Since exp (L/T) = q. We thereupon have established that T is the Sylow p-Subgroup of G and is contained 37 in all components of 0. Now, define T: [U/TzUEGJ. Let g: gTEG/T. Then, g E G implies that there is a component S of 0 such that g E S. Since T E S, it then follows that 'g' = gT E S/T. Let U/T and S/T be distinct elements of 1'. Then U/T fl S/T = U fl S/T = T/T = T and thus, T is a Baer partition of G. Now, U/T ET. is normal in G/T since U is normal in G. Hence all components of T are normal subgroups of G. It then follows from theorem B. 5 that G/T is an elementary abelian q-group which completes the proof. We now wish to say something about the components of cp-partition of G. Due to the combined results of Baer, Kegel, and Suzuki, the structures of all components are essentially determined. To be more specific, Let H be component of a) Baer partition which is not self-normalizing. Then, HCG. If H is normal in G, then H is a preper normal o-admissible sub- group of G. If H is not normal, then let N be the normalizer of H in G. Then, HC N implies that O‘ induces a non-trivial Baer partition 0 on N. Then, H is a proper normal N CTN-admissible subgroup of N. If N is a q-group, then N and hence H is nilpotent which we wish to Show. So, assume that N is not a q-group. It then follows by theorem B. 3 (c) that H. is generated by all those elements of N which do not have order p. 38 So, g E N - H implies that lgl = q. Let h E H. Then, -1 . . -l q ' . . q hg E H 1mplles that (hg- ) =~l. So, thls tOgether w1th g = l -l 2 q- yields that 1 = (hg )q = hhghg ...hg :1 which implies that H admits an liq-automorphism. It then follows by theorem A. 7 that H is nilpotent. This now brings us to our next definition. Definition 2. 2: An automorphism 7 Of order n is said to be an 7 72 V“ Hn-automorphism Of a group G if g. g . g . . . g = l for all g E G. A group G is said to be an Hn-group if G admits an Hn-automorphism. Remark 2. 2.1: A fixed-point-free automorphism of order n of a group G is an Hn-automorphism, but not conversely. The converse, however, is true if n is coprime to lGl. It is still an Open question as to whether a group G which admits a fixed- . point-free automorphism is solvable or not. We new state our result for proper normal o-admissible subgroups of a cp-partition. Theorem 2. 2.1: Let 0' be a normal cp-partition Of a group G . and K a proper normal O-admissible subgroup Of G. Then, there is a prime divisor q of [G:K] such that K is an Hn-group where n divides q exp (0‘), K is a p-group, or the semi-direct product Of a cyclic p~group by a group Of exponent q. Proof: We use induction on lGl. S3 is the ground case. 39 Let gEG - K such that gK has order q in G/K. Define U = . If UCG, then U not contained within a component of 0‘ implies that O induces a normal non-trivial cp-partition 0U on U. Now, K is clearly a proper normal (TU-admissible subgroup of U. Hence, it follows by induction that there is a prime divisor q Of [U:K] such that K is an Hn-group where n divides q exp (0 U). Then, exp (0U) dividing exp (0‘) yields that n divides q exp (0'). SO, let us assume that G = . Now, let g be any element of G - K. Then, G/K being cyclic Of order q implies that gq E K. Since, g E K, there is a component L of 0' such that LE K and g E L. - But, gq E Lfl K a cyclic p-group such that exp (L fl K) divides exp (0). Hence, lgql divides exp (0’). We thereupon have that lgl divides q exp (0'). Assume first that p = q. If K is a p-group, then we. are ‘- through. SO, assume that K is not a p-group. Since lgl divides p exp(d), g is a p-element. Now, CK(g) is a p-group by theorem 2.1.1 (a). Then, K not being a p-group implies that g induces a non-trivial automorphism on K. Assume now that q i5 p. By the previous paragraph lgl divides q exp (:5). Hence CK(g) is the semi-direct product Of a cyclic p-group by a group of exponent q by theOrem 2.1.1 (b). If K is the semi-direct product Of a cyclic p-group'by a group Of exponent q, then we are through. SO, assume to the contrary. Then g induces a non-trivial automo rphism on K. 40 Now, let g E G - K and k E K. Then, kg-l E G - K implies -1 q exp (01 that (kg ) = 1. Let m = q exp (o). It then follows that _1 m m-l m m-l l=(kg ) =kkgkg ...kg g =kkg...kg ;andhence g induces on Hn-automorphism on K where n divides qexp(o), and the proof is complete. Corollary 2. 2. 2: Let 0' be a normal non-trivial cp-partition of a group G and L a component Of O’ which is not self-normalizing. Then, there is a prime divisor q 0f[ NG(L):L] such that L is an Hn-group where n divides q'exp(o). Proof: Since L is not-self-normalizing in G, we have that LCNG(L) = N; hence, N is not contained within a component Of 0'. If N = G, then the result follows directly from theorem 2. 2.1. So, assume that NCG. Then, 0' induces a non-trivial normal cp-partition 0‘ N on N and L is a proper normal 3 -admissible N subgroup of N. Thus, it follows by theorem 2. 2.1 that there is a prime divisor q of [N:L] such that L is an Hn-group where n divides q exp (ON). But, exp (0N) divides exp (0') which yields that n divides q exp (0‘), and the proof is complete. This last theorem reduces to question Of the structure of non- self-normalizing components Of cp—partitions tO that Of the structure Of Hn-groups. CHAPTER III FROBENIUS CP-PARTITIONS In this chapter, we shall generalize the concept of Frobenius partitions to cp-partitions. After establishing a partial generalization of Frobenius's theorem, we shall use this result to obtain information about Frobenius cp-partitions in a limited SCOpe. Baer [4] defines a Frobenius partition of a group G as a normal partition 0 where one of the components, say H, is a. prOper self-normalizing subgroup Of G. By a well-known Frobenius result, theorem A. 6, H has a normal complement in G, say K. K is referred to as the Frobenius kernel Of G and. the complements to K in G are called the Frobenius complements. J . Thompson. [15] established that the Frobenius kernel is always nilpotent. Baer, theorem B. 6, uses Thompson's result to Show that F(G) is the Frobenius kernel of G and that it is a proper O-admissible Hall subgroup Of G. So, the Frobenius complements are Hall sub- groups Of G. The author has not been able to decide whether or not a self-normalizing component of a cp-partition is a Hall subgroup of G. Our discussion will be limited in this respect. Definition 3.1: A normal cp-partition J of a group G is said to 41 42 be a Frobenius cp-partition if one Of the components is a proper self-normalizing subgroup of G. One readily sees that if H is a self-normalizing component of 0, then H has the property that g E G - H =H leg is a cyclic p-group since Hg is a compOnent distinct from H. In order to completely generalize the situation of Frobenius Baer partitions to that of Frobenius cp-partitions, we must at least Show that H has a normal complement in G. However, this is impossible as the following example shows. Example 3.1.1: Consider As, the alternating group on five letters. ~Let O' consist Of the isomorphic copies of A4 in A5 together with the cyclic subgroups Of order 5. A4 has the property that it is a self-normalizing Hall subgroup of A5 and _ fl g . . __ g E A5 A4 =° A4 A4 _ is a cyclic 3 group. However, A4 has no normal-complement in A5 since A5 is simple. We now proceed to Show that Frobenius's theorem can be generalized in a limited Situation. Let Sp be a Sylow p-subgroup of G and Z its center. G is said to be p-normal if ZgE Sp = Zg= Z for all g E G. Lemma 3. 1. 1: Let SP be a Sylow p-subgroup of a group G such - that g E G - NG(SP) =Sp fl Spg is cyclic. Then, G is p-normal. 43 Proof: Assume that G is not p-normal and let Z be the center of a Sylow p-subgroup Of G. Then, there is a g E G such that z i! zgcsp. Let N = NG(Zg). By theorem A. 9, 2g is not normal in SI) and hence, Sp is not contained in N. So, there is a Sylow p-subgroup Sph OfG such that SphE N and SpleESph. It thenfollowsthat S leC-S flShCS le. So, Sle=SflSh. P _ P P " P P P P Since Sph :5 SP, it follows that h E G - N . S . , G( P) Thus 5‘ le = S 05 his cyclic. P P P Now, we have that zg gspn N. It then follows that N fl N C S (Sp )-NS P P cyclic groups are characteristic. But, NS (Zg) = Sp fl N. Hence, P o N (S fl N) = 5 fl N. Now, Zg not normal in S implies that S P p . P P Sp fl N CSP. But, this is contrary to the fact that no proper subgroup (Zg) since Sp fl N is cyclic and all subgroups of Of a p-group is equal to its normalizer. We are, therefore, forced to conclude that G is indeed p-normal, which completes the proof. We now give our partial generalization of Frobe nius's theorem. Theorem 3. 1. 2: Let H be a self-normalizing Hall subgroup of a group G with the property that g E G - H =°H fl Hg is a cyclic p-group, p a fixed prime. Then, H has a normal complement in G if G is solvable or if H is a p-group. 44 Proof: We use induction on lGl. S is the ground case. 3 Case I: CorG(H) if I Let K = CorG(H). Since K is a normal subgroup of G contained in H, we have that KEH fl Hg for all g E G. In particular, if g E G - H, then KEH fl Hg, a cyclic p-group. Choose SEK so that lSl = p. It is clear that S is normal in G and that H/S is a Hall subgroup of 0/5. Let g: gS EG/S - H/S. Then, g EG - H. SO, H/S fl (HIS)? = H/S fl Hg/S = H fl Hg/S, a cyclic p-group. If E = g5 E NclsfH/S). then H/S= H/Sfl (H/S)E = H/S r) Hg/S = H n Hg/S which implies that H = H fl Hg, . that is g E NG(H) = H. Thus, H/S is self—normalizing in G/S. Finally G solvable or H a p-group implies that G/S is solvable or H/S is a p-group. It then follows by induction that H/S has a normal complement, say W/S, in G/S. Now, '1' = W/S n H/S = w a H/S which implies that S = w n H. Also, iHWl/lsl = lHl-lWl/ lelwl Isl = lHl-IWIIISIZ = [H:S] [W25] = lH/Sl lW/Sl = lH/Sl lW/Sl / lH/Sfl W/Sl = lH/S W/Sl = lelsl = [ G:S ] = lGl/lSl. Hence, G = Hw. Now, [WzS] = [WszlW] = [WHzH] = [G:H] . Then, lHl coprime to [ GzH] and SE H imply that lSl is coprime to [WzS] 3 SO, S is aSylow p-Subgroup Of W. If S has anormal complement, say R, in W, then R being a Hall subgroup of W implies that R is characteristic in W. But, W is normal in G. So, 45 R is normal in G. Then, HR = HSR = HW = G. Also, Rfl Hg Wle = S and Rfl S = I yield that Rle = I. Therefore, it suffices to show that S has a normal complement in W. Assume now that H is not a p-group and let h E H have order q i5 p. If w E W so that wh = hw, then h E H fl Hw. But, h is not a p-element. It then follows that w E NG(H) = H and hence, w E W fl H = S. Thus, if h does not centralize S, then h induces a fixed—point-free automorphism of prime order on W. We then have by Thompson's result, theorem A.lO, that W is nil- potent. Then, S being the Sylow p-subgroup of W implies that S has a normal complement in W and we are through. Assume now that h does centralize S and consider H1=< S, h> and G =WH. If glEG -H, thenglEG-Hand l l 1 1 31 ' g . so H fl H C H fl H l, a cyclic p-group. If g E N (H ), then 1 l - 1 G1 1 . g1 g1 HI = H1 fl H1 _C_ H fl H . But, H1 is not a cyclic p-group. Hence, we have that g1 E NG(H) = H. So, g1 E H fl G1 = Hfl WH1 = H1 which yields that H is self-normalizing in G We also have that l 1' [Gle1] = lWH1:Hll = [WM le1] = [w:S] which is coprime to lHl and consequently to lHll. Finally, if HICH, then [G:Gl ] = [WHzWH1 ] = [ H:Hl] >1. It then follows by induction that S has a normal complement in W. So, let us assume that H = H1. Now, H 2 HI = which is abelian. Then HEC = CG(S). 46 But, S normal in G implies that C is normal in G. This together with the fact that H is self-normalizing in G yields that HCC. We assert now that C = G. If C c3G, then we have by induction that H has. a normal complement U in C. Let g e G - C. C normal in G implies that HgE C. 'But, Hg is a f ) complement to ‘U in C. It then follows by the Schur-Zassenhaus I result, theorem A. S, that all complements to U in C are conjugate. So, there is a c E C’ such that Hg = HC. This implies that g c-1E HG(H) = HC‘C. Then, g = (g c-l)c E C, a contradiction to g E G - C. We are thus forced to conclude that C = G. It then follows in particular that S is in the center of W. Using Burnsides resu lt,-theorem A. 12, we are able tO conclude that S has a normal complement in W and thus, H has a normal complement in G. The remaining subcase is where H is-a p-group. But, any normal subgroup Of order p of a p-group is necessarily in the center of the group. So, HC CG(S). By repeating the above arguments we get that C = G and hence, H has a normal complement in G. Case II: CorG(H) = I If G is solvable, then the commutator subgroup G' is proper in G. We assert that HG' = G. Since G' EHG' and G/G' is abelian, it follows that HG' is normal in G. If HG' C3G, then we have by induction that H has a normal complement U in HG'. Let g E C - HG'. Then, HG' normal in G yields that HggHC'. 47 Again, we have by the Schur-Zassenhaus result that there is an t g _ 1.1x -1 .. I x E HG such that H - and hence, gx E NG(H) - HS HG . So, g = (g x-l)x E HG', a contradiction. Thus, we are able to conclude that G = HG'. Let H = H fl G'. It is clear that H is a Hall subgroup Z 2 Of G'. Assume that H2 is not a cyclic p-group. If g E G' - HZ' then g E G. - H and we have that Hzfl HZg E H fl Hg, a cyclic - o E , I: gc g p group If g NG'(H2) then H2 Han2 __Hle . Then, HZ . not a cyclic p-group implies that g E NG(H) = H. So, g E G'fl H = Hz implies that H is self-normalizing in G'. Then, G' C G implies by 2 induction that H2 has a normal complement V in G'. But, V is a Hall subgroup of G' and consequently is normal in G. Then, G = HG' = HHZV = HV, where Hfl V = I. Thus, H has a normal complement in G. Assume now that HZ = Hfl G' is a cyclic p-group. If Hz = I, then G' is a normal complement to H in G. SO, assume that : ° ' : C H2 #1. Now, CorG(H) I implies that N NG(HZ) C. But, G' normal in G yields that H2 = H fl G' is normal in H. So, HEN. If H = N, then NG'(H2) = H which implies by Burnsides 2 theorem that H2 has a normal complement in G', say V. Then, G = HG' = HHZV = HV where H fl V = I. SO, assume that HCN. By induction, it follows that H has a normal complement U in' N. Then, U being the normal p—complement to H in N and H being 2 48 normal in N imply that UECN(HZ). In particular, ' H2 E Z(Nd(HZ) ). Therefore, it follows that H2 has a normal complement M in G' by Burnside's theorem. Finally, we have that G = HG' = HHZ_M= H‘M where H fl M: I. This completes the subcase where G is solvable. Assume now that H is a p-group. We then have by lemma 3.1.1 that G is p-normal. Let Z be the center; of H and N the normalizer in G Of Z. Now, NCG since CorG(H) = I. Also, HE N. If H CN, then it follows by induction that H has a normal complement in Nand hence, H is isomorphic to the largest factor group of N which is a p-group. The same conclusion follows if H = N. But, it follows by the second Hall-Greun theorem, theorem'A. 11, that the largest factor group Of G which is a p-group is isomorphic to the same for N. Therefore, H has a normal complement in G and whence, the proof is complete. The normal complement described in the above theorem is unique when it exists since it is a Hall subgroup of G. This normal complement Shall be referred to as the Frobenius cp-kernel of G. The complements to it in G Shall be called the Frobenius cp-complements. We new state our main result Of this chapter. Theorem 3.1.3: Let G be a Frobenius cp-partition Of a group G where one of the self-normalizing components H is a Hall sub- 49 group. Then, G has a Frobenius cp-kernel K if G is solvable or if H is a p-group. Moreover, if Z(K) contains a p'-element of composite order, then there is a normal component L of 0 such that the following hold: a) KEL; b) [L:K] = 1 iff G is a Frobenius Baer partition; c) If [L:K] > 1, then L/K is a cyclic p-group; d) K is nilpotent unless the Frobenius cp-complements are p-groups. M: Let H be a self-normalizing Hall component of G. Since Frobenius cp-partitions are normal, it follows that Hg is a component of G for all g EG. Then, g E G - H = H i5 H8 since I H is self-normalizing. So, H H Hg is a cyclic p-group for all g E G - H. We then have by theorem 3.1. 2 that G has a Frobenius I cp-kernel, say K, since G is solvable or H is a p-group. Assume now that Z(K) contains a p'-element x of composite order. It then follows by theorem 1. 4. 3 that there is a component L of 0 such that KS L. Since K is normal in G, we have that KS L n Lg for all g E G. But, K is not a cyclic p-group and so, L = L.g for all g E G. Thus, part a is established. Assume now that [L:K] =1. Then, L = K. First, we assert that the elements of H different from 1 induce fired-point- free automorphisms on K. Let h(ié l) E H. If h is neither a 50 p-element nor a p'-element, then it follows by corollary 2.1. 2 that CK(h) EH 0 K = I. So, assume that h is either a p-element or a p'-element.~ If. h is a p-element, then we conclude from theorem 2. l. l (a) that exp(CK(h) ) divides lhl exp (0‘). Since H is a Hall subgroup of G, h(ié l) 6 H is a p-element, and since K is a normal complement to H in G, it follows that K is a. p'-subgroup of G and hence, CK(h) is a p'-subgroup. But, Ih|° exp (0) is a power of p. We then must conclude that CK(h) = I. So, assume that h is a p'-element. Choose n so that Ihn| = q, a prime. Then, C-K(h) E CK(hn). But, lhnl = q where q i5 p implies by theorem 2.1.1 (b) that CK(hn) is the semi-direct product of a subgroup of H 0 K = I by a group of exponent q. Then, lHl and lKl being coprime yield that CK(h) = I. This establishes that CK(h) = I and hence, h induces a fixed-point-free automorphism on K. Our next assertion is that CG(k)EK for all k” l) E K. Let g e CG(k) where kn! 1) E K. Then, G = HK implies that g = hko hk where h E H and k0 E K. We then have that k 0: k or that h l k = kok k0. . So, h induces a fixed-point-free automorphism on K and fixes the conjugate class of K which contains k if h 75 1. Since h induces a fixed-point-free automorphism on K, ko can be h k '1. So, expressed uniquely in the form k0 = k1 1 h h l h h -1 -1 -1 .. _ , k -k1k1 kk1(k1 ) or (k1 kkl) —k1 kkl. But, thlS isa 51 contradiction to the fact that h induces a fixed-point-free automorphism on K. Thus, h = l and hence, g = ko E K as asserted. Now, K is nilpotent by Thompson's result, theorem A. 10. We also have by the Schur-Zassenhaus result, theorem A. 8, that all complements to K in G are conjugate. By theorem A. 13 we conclude K has a complement M such that M 0 Mg = I for all g E G - M. Then, all complements being conjugate implies that M = Hx for some x E G. Hence, all complements have trivial intersection. Now, let L be any component of 0 such that L )5 K. We fir st assert that L n K = I. Assume to the contrary. Then, Lfl K is a cyclic p-group. Now, [L:Lfl K]: [LK:K] . Then, K being a Hall subgroup of K implies that 1 = ( [LKzK] , 'IKI) _ =( [L:Lfl K] , IKI). Also, Ln K is a Hall subgroup of L. Let x E L have order q 35 p. where q divides [L:Lfl K] . Then H being a Hall subgroup of G such that (I Hi, ‘K|) = 1 implies that q divides IH‘ and hence, there is a conjugate HX of H such that x E Hx. It then follows that x 6 Ln I-IX. But x is a p'-element and so, L-= Hx. However, this implies that Hx fl K.= Lfl K i I, a contradiction. So, we are forced to conclude that L E G such that L J5 K=>iLflK = I. Since K = F(G) and Lfl K = I, it follows by theorem B. 6 (c) that there is a conjugate Hy of H such that LE Hy and hence, L = Hy. It now follows that 0‘ is indeed a 52 Frobenius Baer partition of o. This establishes the only if part of b. Assume now that KC L. Then, G = HK implies that L n H i5 I and hence, o is not a Frobenius Baer partition since H and L are components of o. This completes the if part of a. Again, if KC L, then G = HK implies that there is a sub- group H of H such that L = H K. So, L/K = HIK/KZH 1 1 1' But HI = L 0 H a cyclic p-group since L and H are distinct components. Finally, if H is not a p-group, then let h E H have order q f p. We have by theorem 2.1.1(b) that CK(h) is the semi-direct product of a subgroup of H H L by a group of exponent q. But, lHl and IKI being coprime implies that CK(h) = I. Thus, h induces a fixed-point-free automorphism of prime order on K and so, K is nilpotent by theorem A. 10. The theorem is now proved. The conclusion that K is contained in a component of o is false if Z(K) does not contain an element of composite order as the following example shows. 5 Example 3. 1. 2. Let K = < a,b: a5 = b = l >. Consider the group of automorphisms of K defined by H = =. This is theorem 8. 4.1 in M. Hall's Group Theory. Let (p be an automorphism of a group G and H a subgroup, then H is said to be Cp-invariant if Hcp = H. If H is a normal cp-invariant subgroup of G, then cp induces an automorphism E cp on G/H in the following natural fashion: $(g H) = g H. Before we state the next theorem, an automorphism (p of G is said to be fixed-point-free if gCP = g implies that g = 1. Theorem A. 4: Let (p be a fixed-point-free automorphism of G and H a normal cp-invariant subgroup of G. Then, (p induces a fixed-point-free automorphism on G/H. 54 55 The statement of this theorem is given in Schenkman, page 279, and its proof is left as an exercise. Now, if S I is a non-empty set and if G is a group, then G is said to be an S-group if there is a function * from G x S into C such that (g1g2)* s = (gl* 5) (g2* s) for all g1, g2 E G. s E S. A particular case of this is where S is a group of automorphisms of G. Two S-groups G1 and G2 are said to be S-isomorphic if there is an isomorphism (p from G1 onto G2 such that CP(g * s) = cp(g) * s. ' Now, a subgroup H of an S-group G is said to be an S-subgroup if h *_s_E H for all h E H. Theorem A. 5: Let G be an S-group and KSHSG normal S-subgroups of G. Then, the relation U = {(g H, gK(H/K) )‘g EG } is ants-isomorphism of G/H onto G/K/ H/K. This is theorem 2. 9. 4 in W. Scott and its proof is left as an exericse. Theorem A. 6: Let G be a finite group and H a subgroup of G such that g E G - H implies that Hg H H = I. Then, H has a normal complement in G. This is a well—known Frobenius theorem-~see W. Scott, theorem 12. 5.11. The normal complement to H is referred to as the Frobenius kernel of G and has been shown to be nilpotent by Thompson [15] . 56 Now, a group G is said to be an Hq-group if it admits an 7 72 7‘14 automorphism y of order q such that g-g .g g =1 for all g EG. Theorem A. 7: All Hq-groups are nilpotent. This theorem was proved by O. Kegel, Math A. 75, 373-376 (1961). A subgroup H of G is said to be Hall if lHl and [G:H] are c0prime. Theorem A. 8: If. K is a normal Hall subgroup of a group G such that K or G [K is solvable, then any two complements of K are conjugate. This result was proved by H. Zassenhaus and is theorem 9. 3. 9 in W.‘ Scott's book. Theorem A. 9: If P1 and P2 are Sylow p-subgroups of G such that , then Z(Pl) == Z(PZ). Z(Pl) is normal in P2 This is theorem l3. 5. 3 in W. Scott's book. An automorphism ‘>’ of a group G is said to be fixed-point- free if gyz g implies that g = 1. Theorem A. 10: Any group G which admits a fixed—point-free automorphism of prime order is nilpotent. 57 This classic result was first proved by Thompson [15] , and its proof may be found in Schenkman, theorem IX. 4. h. Theorem A. 11: If G is p-normal, then the largest factor group of G which is a p-group is isomorphic to the same for the normalizer of the center of a Sylow p-subgroup. This is known as the second theorem of Hall-Gran and is theorem 14. 4. 6 in M. Hall's book. Theorem A. 12: If P is an abelian Sylow p-subgroup of G which , is in the center of its normalizer, then P has a normal complement in G. This is known as Burnside's theorem and is theorem 14. 3.1 in M. Hall' 5 Book. W. Scott in his book defines a Frobenius group as one which contains a proper normal subgroup K (called the Frobenius kernel) such that if k(9‘- l) E K, then CG(k)EK, see page 348. He then proves the following theorem, theorem 12. 6.1. Theorem A. 13: If G is a Frobenius group with Frobenius kernel K, then there is a Hall subgroup H such that G = HK and Han=1£ora11geo-H. APPENDIX B Theorem B.1: If 0 is a Baer partition of a group G and a, b E G are commutable elements of G, then a and b are contained within a common component unless they both have order q for some prime q. For the proof of this theorem see Baer[ 4] , lemma 2.1. Theorem B. 2: If G is a normal non-trivial Baer partition of a non-simple group G, then the following are equivalent: a) o is simple - b) If F(G) ‘# I, then G is isomorphic with S4 c) If G is not isomorphic with S then S(G) is simple and 4’ non-abelian. d) G contains S as a subgroup. 4 For the proof, see Baer [2] , page 2. Theorem B. 3: If G is a normal Baer partition and K a proper normal O-admissible subgroup of G, then the following hold: a) o is not Frobenius if and only if [G:K] and lKl are not coprime. b) If G is not Frobenius, then there is a prime p such that g E o - K implies lgl = p. c) If 0' is not Frobenius and G is not a p-group, then [G:K] = p 58 59 where p is a prime. Also, K and G both are extensions of nilpotent groups by a p-group. Finally, K is a component which is generated by all g E G such that lgl # p. For the proof, see Baer [4] , theorem 5.1. Theorem B.4: A p-group G where (G) #p admits anon-trivial partition if Hq(G) C G. This was proved by O. Kegel [8]. Theorem B. 5: If G is a non-trivial Baer partition of a group G and each component of o is a normal subgroup of G, then G is an elementary abelian q-group. For the proof of this theorem see 0. Kegel [8] , pages 172-173. ' Theorem B. 6: Every Frobenius partition of a group G has the following properties. a) F(G) is a proper normal o—admissible Hall subgroup of G. b) All self—normalizing components are conjugate. c) A subgroup U of G is contained within a self-normalizing component if and only if U f) F (G) = I. (1) Every normal subgroup not contained in F(G) contains F(G); and the proper normal o-admissible subgroups are contained. in F(G). e) The self-normalizing components faithfully induce on F(G) 60 Frobenius groups of automorphisms and do not contain any 2 elementary abelian subgroups of order p . f) If a subgroup U of G contains a self-normalizing component, then U is self-normalizing. This was proved by Baer [4] , theorem 4.1. Theorem B. 7:. If the sockel S(G) of a finite group is neither abelian nor simple, then any normal partition on G is trivial. - This was proved by Baer [4] , theorem 3. 6. Theorem B. 8: If G is a normal non-trivial partition of a finite group G and F(G) :5 I, o is then and only then simple when the following occur: a) G is isomorphic with S4; b) S(G) is elementary abelian of order 4; c) 0 consists precisely of those cyclic subgroups of G which are not contained in S(G). This is a result of Baer [l l , theorem A. Theorem B. 9: Let 0 be a non-trivial partition of a finite group G and assume that S(G) is non-abelian simple. Then, the following hold: a) [G:S(G)] = '2 b) The Sylow p-subgroups of odd order are abelian c) The Sylow Z-subgroups are D-groups, that is, non-abelian groups 61 which contain an abelian subgroup H of index 2 and an involution go outside of H such that (goh)z= 1 for all h E H. This was proved by Baer [2] , Hauptsatz, page 1. INDEX OF NOTATION 1. Relations and Sets _C_.' Is a subset of C Is a proper subset of 5 Is less than or equal to < Is less than E Is an element of II. Groups I The trivial group S(G) = < H:H # I is a minimal normal subgroup of G> F(G) = <_H:H is nilpotent normal subgroup of G > GL(n, q) = The) group of n x n non-singular matrices over GF(q) ' PGL(n, q) = GL(n, q)/Z(GL(n. q) ") SL(n, q) = (A E GL(n, q): det A1=l] PSL(n, q) =‘ SL(n, q) / Z(SL(n, q) ) IGl the-order of G (g) the order of g 11 a set of primes n [n] the set of all prime divisors of n: l. fi-subgroup 11 [II-1‘ ] C: n 62 - 63 TT-element n[|gi] in n'-subgroup n [lHl] 0 TT :Q n'-element “(lgllnn =0 h8 The image of h under 6 H8 The image of H under 8 I(G) The inner automorphism group of G . exp(G) = smallest positive integer n such that gn: lufor all g E G < —,. . .> The group generated by . . . Z(G) _ Hn(G) = h(G) _ Largest element of the ascending central series of G. CK(H) = {k e K:kh = hk for all h E H } CK(h) = {k E K:kh = hk } y-invariant Hy: H 7 is fixed-point-free g7 = g = g = l (3),: {gEngy=g] k NK(L)= {kELzL =L} -1 -1 (x. Y] = x y xv T c:,'. ’ .h.-I". You. 64 . n-l Hn-automorphism Iyl = n and g gy- g . . . g‘y = l for all g E G H is Hall in G lHl is coprime to [G:H] p-normal Z(S )gCS =Z(S )g = Z(S ) for all g EG P " P P P I = < . Cor (H) = n Hg G gEG l. 10. 11. 12. 13. BIBLIOGRAPHY Baer, R. "Einfache Partitionen endlicher Gruppen mit nicht- trivialer Fittingscher Untergruppe, " Arch. Der Mathematik XII, (1961), 81-99. "Einfache Partitionen nicht-einfacher Gruppen, " Math. Z. LXXVII, (1961), 1-37. __________ "Group Elements of Prime Power Order. " Am. Math. Soc. Trans. LXXV, (1953), 20-47. "Partitionen enlicher Gruppen, " Math. Z. LXXVH' (1961), 33-376. Hall, M. Jr. . The Theory of Groups. New York: Macmillan, 1959. Hughes D. and Thompson J. "The H -Prob1em and the Structure of the H -Groups, " p Pac. J. of Math. IX, (1959). 1097-1102. P - Iwahori, N. and Kondo, T. "A Criterion'for the Existence of a Non-Trivial Partition of a Finite Group with Applications to Finite Reflection Groups, " J. Math. Soc. 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