On an Embedding Property of Generalized Carter Subgroups Citation Cline, Edward Thomas (1966) On an Embedding Property of Generalized Carter Subgroups. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/S8GH-XA56. https://resolver.caltech.edu/CaltechTHESIS:09222015-083434916 Abstract If E and F are saturated formations, we say that E is strongly contained in F if for any solvable group G with E -subgroup, E, and F -subgroup, F, some conjugate of E is contained in F. In this paper, we investigate the problem of finding the formations which strongly contain a fixed saturated formation E . Our main results are restricted to formations, E , such that E = {G|G/F(G) ϵ T }, where T is a non-empty formation of solvable groups, and F(G) is the Fitting subgroup of G. If T consists only of the identity, then E = N , the class of nilpotent groups, and for any solvable group, G, the N -subgroups of G are the Carter subgroups of G. We give a characterization of strong containment which depends only on the formations E , and F . From this characterization, we prove: If T is a non-empty formation of solvable groups, E = {G|G/F(G) ϵ T }, and E is strongly contained in F , then (1) there is a formation V such that F = {G|G/F(G) ϵ V }. (2) If for each prime p, we assume that T does not contain the class, S p’ , of all solvable p’-groups, then either E = F , or F contains all solvable groups. This solves the problem for the Carter subgroups. We prove the following result to show that the hypothesis of (2) is not redundant: If R = {G|G/F(G) ϵ S r’ }, then there are infinitely many formations which strongly contain R . Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Mathematics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Hall, Marshall Thesis Committee: Unknown, Unknown Defense Date: 1 January 1966 Funders: Funding Agency Grant Number Caltech UNSPECIFIED NSF UNSPECIFIED ARCS Foundation UNSPECIFIED Record Number: CaltechTHESIS:09222015-083434916 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:09222015-083434916 DOI: 10.7907/S8GH-XA56 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 9163 Collection: CaltechTHESIS Deposited By: INVALID USER Deposited On: 22 Sep 2015 19:48 Last Modified: 27 Feb 2024 22:05 Thesis Files Preview PDF - Final Version See Usage Policy. 13MB Repository Staff Only: item control page

ON AN EMBEDDING PROPERTY OF GENERALIZED CARTER SUIDROUPS Thesis By F.dward T. Cline, Jr. In Partial Fulf'illment of the Requirements For the Degree of Doctor of Philosophy California Institute of Technology Pasadena, Cali.fornia 1966 -ii- ACKNOWLEDGMENTS · I would like to thank Professor Marshall Hall, Jr. for his help and encouragement in the preparation of this the.s is. In addition to this, I am grateful for his instruction throughout both my undergraduate and graduate s~udy. I have also benefited from many discussions with Professor E. c. Dade, and am thankful for his proof to Lemma.?. Finally, I am grateful to the California Institute of Technology, the National Science Foundation, and the ARCS Foundation, for financial assistance. -iii- AmTRACT If' eand "!are saturated :formations, we say that eis strongly contained in 1 i:f for any solvable group G with ~-subgroup, E, and 1-subgroup, F, some conjugate o:f E is contained in F. In this paper, we investigate the problem of finding the formations which strongly contain a fixed saturated formation e. Our main results are restricted to :formations,e, such that ~ = [Gj G/F(G) E'f'}, where 1 is a non-empty formation of solvable groups, and F(G) is the Fitting subgroup of G. of the identity' then ~ Ir"r consists only =n' the class of nilpotent groups' and for any solvable group, G, the 'n-subgroups of G are the Carter subgroups of G. We give a characterization of strong containment which depends only on the f'ormations e' and 1. . From this characterization, we prove: If~ is a non-empty formation of solvable groups, i = {GjG/F(G) ET}, and 2 is strongly contained in '1-, then (1) there is a formation 4.1 such that '1= fGIG/F(G) E1J}. (2) If for each prime p, we assume that 1' does not contain the class, i p ,, of all solvable p'-groups, then either ~ =1, or 1 contains all ' solvable groups. This solves the problem for the Carter subgroups. -iv- We prove the following result to show that the hypothesis of (2) is not redundant: I f (R= (GIG/F(G) E ~r 1 }, then there are infinitely many for- mations which strongly contain ~. -1- Chapter I Introduction In 1961, Roger Carter [2] discovered the interesting result that a .finite solvable group contains a nilpotent subgroup which is its own normalizer. groups. These subgroups are now called Carter sub- In the same paper, Carter showed that a:ny two Carter sub- groups are conjugate, and that they possess certain interesting cover-avoidance properties. Since the discovery of the Carter subgroups, several papers, devoted to the investigation of relations between the system normalizers of a finite solvable group and its Carter subgroups, have been published. In another direction, Gaschtttz [5] has discovered a quite general theory from which most of the properties of Carter subgroups follow as a special case. One consequence of this theory is that there are many types of subgroups of a finite solvable group, G, which possess properties analogous to the properties of Carter subgroups; a subgroup of G which satisfies the necessary conditions is called a generalized Carter subgroup. We shall, f'rom now on, assume that all groups under discussion are finite and solvable. Definition 1. satisfies: To describe our results we need A formation, ""f, is a collection of groups which -2- (1) If G E 1, and N is norma1 in G, then G/N E 1. 2 are norma1 _i n G, and G/Ni E 1, then G/N1 n N2 E 1. A formation 1 is saturated i f it satisfies, in addition to (1) and (2) I f N1 ,N (2), (3) I f G/D(G) E1, and D(G) is the Frattini subgroup of G, then G E 1. The generalized Carter subgroups may be defined as foll.ows: Definition 2. Let G be a group, and "f. a formation. A subgroup, F, of G is called an 1-subgroup of G provided (4) F E "J (5) I f U is any subgroup of G containing F, and i f N is a norma1 subgroup of_U such that U/N E °.J, then FN = U (i.e., F covers U/N). One of the basic results on formations is that if ' i s any saturated formation, and if G is any solvable group, then 1-subgroups of G always exist, and all. such groups are conjugate (Gaschtttz, [5]). In this thesis we shall investigate the behaviour of the e-subgroups of a solvable group G with respect to the 1-subgroups of G for two saturated formations e,and 1. Definition J. Let ~ and "J be saturated formations. i is strongly contained in 1 (written ~<< '1) provided (6) If G is any solvable group which has £-subgroup, E, and 1-sub- group, F, than soma conjugate of E is contained in F. We shall be concerned with the problem: Given a saturated formation -3- i, find all formations 1 such that e<< 1. From the definition, it is clear that there are always two formations, °f., which strongly contain~, viz. e, and..8, the formation of all solvable groups. We comment that the relation << forms a partial ordering of the collection of all saturated formations. the definition that if e<< 1, then It is a consequence of eis contained in 1. The two main results of this thesis appear in Chapter V. The first of these shows that if "f is a non-empty formation, and 2 = (GIG/F(G) E<7f} (F(G) is the Fitting subgroup of G), then a:ny formation '1 which strongly contains f!. has essentially the same structure as ein that there is a non-empty formation 'V such that 1 = (GI G/F(G) E lJJ. We then appl.y this result to find a large class of formations which are maximal in the partial ordering << • In particular, if ~ denotes the formation of groups of nilpotent length at most i, then 'Ii is maximal in .the ordering<<. It is easily shown that for 'h = \ , the 'n-subgroups of a group G are the Carter subgroups of G, so the problem is solved for the Carter subgroups. The machinery for proving these results is developed in Chapter r.v. We begin by developing a characterization of strong containment which depends only on the two f'orma.tions . characterization depends on the knowledge that if' e and 1. This eis a saturated formation, then ~ is a locally defined formation (s~e Chapter II), a result proved by Lubeseder in [10]. In certain cases, we are able -4- to strengthen our characterization of strong containment so that it gives a complete description of the minimal local definition for ~ as a necessary condition for strong containment. In Chapter VI, we present an example which shows that Hypothesis II of our main theorem is not redundant. The formation which gives the example is <M. = fGI G/F(G) is a.n r•-group}. It is apparent from Theorem 11 that (R, is not maximal in the partial ordering<<. In fact, there is . an infinite number of formations which strongly contain lX. In Chapter III, we give some preliminary results which are used throughout the paper. In particular, we give a cover-avoidance characterization of the 1-subgroups of a solvable group. In Theorem 5, we find the'1-subgroups of a certain type of semi-direct product. We also show how these subgroups are embedded in the product.; The remaining results in Chapter III are known, but their proofs are included for the sake of completeness. In Chapter II, we give notation along with the results quoted .from previous papers. Some of the results in this chapter are proved because we have found no suitable reference to them in the literature. -5- Chapter II Notation and Quoted Results We shall use the following notation: G - A finite solvable group !GI - The order of G H .S: G - H is a subgroup of G (proper i f H < G) H <l G - H is normal. in G c - A subset of (proper subset if c) D(G) - The Frattini subgroup of G, the intersection of all maximal subgroups of G F(G) - The Fitting subgroup of G, the maximal. normal nilpotent subgroup of G F (G) - The inverse image in G of F(G/F _1 (G)) 1 1 Aut(G) - The automorphism group of G E - A field Zp - The field of integers mod p, p a. prime E(G) - The group ring of G over the field E ::!. E(G) - Is E(G) - isomorphic to TI - A set of primes n' - The complementary set of primes O (G) - The maximal norma1 TI-subgroup of G TI -6- 0 · (G) - The inverse ima.ge in G of 0 (G/O (G)) TT1' • • • 'TTn rrn TT1' • • • ,rrn-1 . OTT(G) - The smallest normal subgroup N of G such that G/N is a n-group TT , ••• ,TTn 0 1 TT TT , ••• ,TT 1 n- ( G) ) • ( G) - 0 n( 0 1. I f H ~ K, and H~ K ~ G, then K/H is a section of G. · a TT-section if K/H is a TT-group. K/H is I f L~ G, then L covers the section K/H provided (L n K)H = K, and avoids K/H if K n L :::_ H. If F acts on G (i.e., every element of F induces an automor- phism of G), and F fixes both H and K, then K/H is an F-invariant section of G. If F acts irreducibly on K/H, then K/H is an F-composi- tion factor of G. An F-composition series of G is a series, G = G0 > G1 ·> ••• > Gn = 1,such that G1 /Gi+i is an F-composition factor of G for each i. The F-composition length of G is n, the length of any F-composition series of G. = G, then K/H is a chief factor of G, and the series, G = G0 > G1 > ••• > Gn = 1,is a chief s eries of G. If F If K/H is an F-invariant section of G, then CF(K/H) denotes the kernel of the representation of F on K/H. CK/H(F) denotes the elements of K/H fixed by every element of F. If F acts on G, and K/H is an F-invariant sect ion of G which is also an elementary abelian p-group, then we may look at K/H as a vector space over Zp , hence also as a Zp (F)-module with the action -7(Hk):r = ml for f E F, Hk E K/H. I:f M is a right Z (G)-module, we may :form the semi-direct p . product GM, where the action o:f G on Mby conjugation is :for m E M, g E G. We shall frequently switch back and forth between additive and multiplicative notation. I:f M and N are right E ( G)-modules, then M ~ N will denote the direct sum o:f these modules, and Hom E(G)(M,N) will denote the collection of all E(G)-linear homomorphisms from M into N. I:f H is a second group, and T is a E(G)-module, then the tensor product M ®ET (sometimes written M ® T) is a E(G X H)-module under the canonical action (m® t)(g x h) =mg® th A principal indecomposable E(G)-module is an indecomposable component of the regular E(G)-module. I:f M is a completely reducible E(G)-module, and M = I 1 @ . . •~Is is a decomposition of M as a direct sum of irreducible submodules, ~' then for each k we let Mr be k the direct sum of all copies of Ik which appear in this decomposition of M. ... ~ is called a prima~ component o:f M. ~ The dual E(G)-module to M will be denoted by M. We recall tha.t M is the collection of all E-linear f'unctionals from M into E, and tha.t Mbecomes a right E(G)-module i f we let G operate on ~ so that -8- ... for g in G, and µ in M, µg is the functional defined by for m E M If N is a ~(G)-submodul.e of M, we set *1- = (µ E Ml(N)µ = O}; we note that the following relations hold if we assume M is finitely gener- ated and idantify M with its second dual. (7) (~)1 = N, (o)l = M, vJ. = (o) If N, U are ~(G)-submodul.es of M such that N s; U, then (8) ~JJ~ E(G)(WN), and (9) CG(U/N) J s; J, ~ Furthermore, i f I 1 and I 2 = CG((U/N)). are submodules of M, then (10) Suppose H <t G, and U is a E(H)-module. For a fixed g in G, we can define a new E(H)-modul.e Ug by looking at U as a E-vector space and specif'ying a new action for H, for h E H The module Ug is called the conjugate of U by g. If F,::; G, and Mis a E(G)-modul.e, then MjF denotes the S(F)-modul.e obtained from M by restriction to E(F). I f U is a E(F)-module, then ujG is the E:(G)-module induced from u. One of the most convenient ways to define UIG is presented in Green's paper [8]. -9- Set UIG = U @E(F)E(G), then UjG can be written as a direct Let 1 = g1 ,g ,•••,g be 2 8 a complete set of coset representatives for F in G. Then sum of E-subspaces in the following way: This decomposition does not depend on the choice of coset representatives. The action of G on UIG is characterized by giving the action of G on each element m © gi: (11) (m ® g.J. )g = mfg ® gj where g.g = f g gj' J. and f g E F~ Using (11), one can show that U is canonically embedded as a submodule of UIGIF by the mapping (12) 6: u-+ u ® 1 for u E u. We shall use the following results: (13) ([1], p.8) Let M = I 1 E9 ••• ~In be a decomposition of the E(G)-module M into a direct sum of irreducible g(G)-submodules. Let ~ be the primary component of M associated with Iit• If w is a:ny g(G)-submodule of M isomorphic to lie' then Ws; ~· (14) ([4], p.75) I:f' Eis a £ield of characteristic p, G is a group, and M is an irreducible E(G)-modul.e, then Op(G) lies in the kernel of the representation of G on M. -10- Let H be a p'-subgroup of G, and let M be the z (G)-module (15) p which gives the permutation representation of G on the cosets of H, 1 ~ • • • @ Jn is any decomposition of M into a direct sum 0£ indecomposable Zp(G)-modules, then each Jk is a principa1 1 HI G. If M = J indecomposable Zp (G)-module. . Proof. I f R is the regular Zp(H)-module, then RIG is the regular Z (G)-module. p Since H is a p'-group, R is completely reducible to a direct sum of Zp(H)-irreducible submodules, one of which is the identity Z (H)-module, I. p R =I IPL. We have, therefore, a decomposition Thus RIG= IIG e LIG, so because M = IIG, each Jk is a direct summand of the regular Z (G)-module. p (16) ([3], p. 372) Let G be a group, and J a principal indecomposable E(G)-module for some field E. Then J has a unique maximal proper E(G)-submodule which we shall denote by/J. (17) (cf. [7], Theorem 1) Suppose A is a group of automorphisms of the group G, and C!Aj,jGj) = 1. If H/K is an A-invariant section of G, then CG(A) covers CH/K(A). The next three results are well-known elementary facts that prove to be quite useful. Their proofs will be included in an appen- dix. (18') I f N ~ G, and n ,•••,nn are sets of primes, then 1 -11- (18) Let G be a group, then 0p , ,p (G) = n CG(K), where the inter- section is taken over all p-chief factors , K, of G. (19) Let G be a group, then F(G) = n Op , ,p (G), where the intersec- tion is taken over all primes p. We shall give a short summary of the theory of formations which is found in the papers of Gaschtttz, and Uibeseder ([5],[6], and [10]). Let ~ be the collection of all solvable groups. Definition 4. For each prime p, let ';/- be a formation. p (a) If "J = Let t p is non-empty, and K is a p-ohief factor of G, then G/CG(K) E "J.P. G E,.& (~) If ~ is empty, then G is a p'-group. p dJ'. is a formation which is called the formation locally defined by the family {e7J.p}. In gener~, a formation,'), is locally defined if there exists a family, (''JP}, of formations such that °:f is locally defined by the family {1p). Because of (18) and the fact that '1 is a formation, (2) is p equivalent to the condition: (a') If "J.p is non-empty, then G/ 0 p , ,p( G) E 1.p • The :family, {1p}, of :formations which locally defines ~ is usually not unique. If, however, {".f ) and {'.ft ) are two families p p of formations which locally define the same formation1, then the .family {1{ IW = 1. n 1' ) also de.fines '1. p"p p p Thus there is a unique -12- minimal local definition for any locally defined formation "!. For example, the minimal local definition of the formation of all p-groups, '?{, is ;i-t p = (1} '/-{ q = empty if q '/:. p (20) ([5], p. 302; [6], p. 198) If 7 is locally defined, then 1 is saturated. (21) ([10]) If tis saturated, then1 is locally defined. From now on, we shall use the terms saturated and locally defined interchangeably. Definition 5. Cl.= 'f If '1 is a formation, and G is a group, then n N whero f = {Nj N <l G, and G/N E 1). NE~ Using this definition, a subgroup, F, of G is an 1-subgroup of G if, and only if, 1, and (22) F E (2'.3) i f F $. U $. G, then F covers U/U'f. (24) ([5], p. '.301) If the formation, vable group G has an 1-subgroup. (25) ([5], P. J01) an element of (a) of U. Let 1, is saturated, then every sol- All the 1-subgroups of G are conjugate. 1 be a formation, and G a group. Let F be 1 such that F $. G. Then If F is an 1-subgroup of G, and F $. U $. G, then F is an 1-subgroup -13- (b) If N <J G, and F is an 1-subgroup of G, then FN/N is an 1-subgroup of G/N. (c) I:f N <J G, F /N is an 1-subgroup of G/N, and F is an 1-subgroup 1 of F , then F is an 1-subgroup of G. 1 -14Chapter III Preliminary Results The first theorem of this chapter is a result similar to Clifford's theorem. Theorem 1. Let H be a normal p'-subgroup of G, E a field of characteristic p, and Man indecomposable E(G)-module. Then MIH is a completely reducible S(H)-module whose irreducible components form a single class, C, of conjugate E(H)-modules under conjugation by elements of G. Let L, J be two E(G)-submodules of M such that Le J. Then the distinct E(H)-irreducible components of (J/L)IH are precisely the elements of c. Proof. The complete reducibility of MIH follows directly from the theorem of complete reducibility. Let MIH = I 1 ~ ••• e Is be a decomposition of M as a direct sum of irreducible E(H)-submodules, and for each k = 1,•••,s, let~ be the primary component of MIR associated with Ik. Consider the module Ik g for some g in G. The action of H on this module is the same as the action of H on the conjugate module ~ because of the formula: 1 (i0)h = ik(ghg- )g • Now I k·g Clearly is isomorphic to some Ij, hence ·by (13), I k g I j g -l is isomorphic to Ik' so I j g -l S ~· s:; Mj. Therefore -15~ := Mk g = Mj, so G permutes the primary components of Ml H by conjugation. Let c1 ,•••,Cr be the conjugacy classes of primary components of Ml H under the action of G. Set Ti = l ~Mic• Because Ml H is MicE Ci the direct sum of its primary components, Ml H = T1 ~ ••• ~Tr• But each Ti is a E(G)-submodule of M, so we contradict the indecomposability of M unless r = 1. Let C be the conjugacy class of irreducible components of MjH. Obviously, all of the irreducible E(H)-submodules of (J/L)IH lie in c. Since G is transitive on the elements of C, and J/L is a E(G)-module (non-zero), every element of C must appear as a component of (J/L)jH. Lennna 1. Let G be solvable, and M be a Z (G)-module. p M is :f'aith:f'u.l if, and only i:f',' MIF(G) is faith:f'u.l. Proof. The lemma follows a fortiori from the statement that 1:f' 1 < N ~ G, then 1 < N n F(G). Now 1 < N, and N is solvable, so 1 <.F(N) = N n F(G), so the lemma holds. The second theorem in this chapter is the "Frobenius reciprocity theorem" for modules. paper. It is one of the main tools used in this For ordinary representations, the usua1 reciprocity theorem may be derived from this isomorphism by noting that 1:f' µ., and Tl are tha characters of two E(G)-modules, M and N, then the scalar product -16(~,1)) is the E-dimension of Hom E(G)(M,N). Theorem 2. Let G be a group, and H S G. and Na E(G)-module. Let M be a ?.(H)-module, Then Hom E(G)(MIG,N), and Hom E(H)(M,NjH) are ·isomorphic as vector spaces over E. Proof. We define a map El, from Holll,s(G)(MIG,N) into Hom E(H)(M,NjH) by where A is the cn.nonical map defined by (12). defined. El is clearl.y well If' m E M, and h E H, then [mh]®(1)) = [mh ® 1J111 M® 1 = [ (m © 1 )hJTJI M ® 1 = {[ m ® 1J11IM ® 1}h = ([m]El(Tj)}h • In this computation, we have used (11) and the fact that TJIM ® is an element of Hom_E(H)(M ® 1,N). 1 Both t:. and 11IM ® 1 are E-linear maps, so e(T)) is a E(H)-:homomorphism from M into NIH • Since restric- tion is a E-linear operation, it follows that El is a :?:-linear map from Hom E(G)(MjG,N) into Hom E(H)(M,NIH). Suppose ®(11) = o. Then 11IM@ 1 = O, so using (11) we have (m@ g.i)1) = ((m ® 1)g_e)il = {(m ® 1)T))g; = Og,{ . = O, since 1) is a E(G)-homomorphism. Therefore ® is a 1-1 map. Letµ be an arbitrary element of Hom E(H)(M,NjH). We set -17s s ~=1 1=1 (l (mA ® g' ))µ,* = l (m;)µ,gi, where g1 = 1, g2 ,•••,g are a set 5 of coset representatives for H in G, and [G:H] = s. Since N is a E(G)-moduJ.e, µ* is a well defined map from MIG into N. s g E G, and v = l m~ ® g~ lies in MIG, then J=1 s (vg)µ.* ZS rl s 1 = {I (me @ g, )g µ.• ~=1 gy(~ ,}µ.• l Cm,thg)µ~a > = l (m,e )µ.hggy(~) )=1 s = m,1hg ® i=1 s s = If [I 1=1 cm,)µ.g.e]g = (v)µ•g ' R=1 where y is the permutation of the cosets Hg1 induced by multiplication on the right by g. We have used the fact that µ. is a ~(H)-homomor- phism, as well as equation (11). Becauseµ. is :;:-linear, µ,• is E-linear, henceµ• lies in Hom E(G)CMIG,N). I f m E M, then [m]®(µ*) = (m ® 1)µ•j M ® 1 = · (m)jAo ~ therefore ®(µ*) = µ. = (m)µ., This shows that e maps Hom E(G)(MjG,N) onto Hom E(H)(M,NjH), and completes the proof. We shall begin a discussion of the properties of 1'-subgroups of solvable ·groups. It is useful. to know the behavior of G7 (cf. Def. 5) under homomorphisms, so we prove -18- Let 1 be a formation, G a group, and H a normal subgroup Lemma 2. of G. Then (G/H)"J = "'JH/H Proof. Let F be the inverse image, in G, of (G/H)'J. Then G/F is isomorphic to (G/H)/(G/H)1 , hence G/F lies in 1. Therefore, GyHs;F. Since G1 $ G1H, G/G1H is an element of is isomorphic to (G/H)/(G1H/H). have F $: ~H. 1. Furthermore, G/G1H Therefore F/H s; G1H/H, so we also This shows that F = G1H, and completes the proof. The next two theorems give cover-avoidance properties of 1-subgroups. They generalize a remark made by Carter in [2]. Theorem 3. Let 1 be a formation locally defined by the family £1p }, G be a group, F a subgroup of G which lies in 'f, and K an F-composition factor of G. Then (a) F either covers, or avoids K; (b) if F covers K, and pl jKj, then F/CF(K) E 1~; (c) i f Fis an 1-subgroup of G, and .pl IKI, then (C) F/CF(K) E Proof. Let K 1P -=-> F covers K. = L/M be the F-composition factor in question. Statement (a) follows from the fact that F acts irreducibly on K, and (L n F)M/M is an F-invariant subgroup of K. -19If F covers K, then L/M = ( L n F)M/M. If we look at F as a 1 set of operators on K, then K is operator isomorphic to L n F/M n F, a p-chief fantor of F. Therefore the kernel of the representation of' F on L n F/M n F is CF(K). F/CF(K) E "JP. Because F lies in'/, it :f'ollows that This proves (b). Now suppose F is an 1-subgroup of' G, and K is a p-section of · ' G such that F/CF(K) lies in 1 . To show F covers K, it suffices p to show that F covers the larger section FL/M. But by (25a), F is an 1-subgroup of FL, hence it is sufficient to show F = FL/M is an element o:f' 1 since F, by definition, covers any such factor of FL. If q # p, then K, as a normal q'-subgroup of F, is contained in 0q , (F). Therefore 0q , ,q (F)L/M S 0q , ,q (F), so F/Oq , ,q(F) is isomorphic to a quotient group of FL/Oq , ,q(F)L. FL/Oq , ,q(F)L is -isomorphic to F/Oq , ,q(F)(F n L), a factor of F/Oq , ,q(F). Since F E 1, F/Oq , ,q(F) E 1q , hence F/Oq , ,q (F) is also an element of 1_. q Let U = Ferr • Since F E f, F/O , (F) E '1 ·, hence U < O , (F). 7p p ,p p - p ,p Since F/CF(K) E 1p' U $_ CF(K), hence K is contained in the center of UL/M. Therefore UL/M has a normal p-complement. By (18'), UL/Mis contained in 0, p ,p(F) p ,p(F), thus F/O, is a factor grou.p of' FL/UL~ F/U(F n L). E Therefore, F/Op •,p(F) locally defined, F lies in 1-. 1p. Because "J is -20- Our next. theorem will show that property (C) the 1-subgroups of a solvable group G. characterizes In order to obtain as weak an hypothesis as possible, we prove two lemmas. Property (C) actually applies only to specific F-composition factors of G, so when we say that property (C) holds for an F-composition series, G = G0 > G1 > ••• > Gn = 1, of G, we mean that F satisfies this property for all factors, Gi/Gi+i' of the series for which the hypothesis of property (C) holds. Lemma 3. and F ~ G. Suppose 1 is a formation locally defined by {1P), F E 1, Let A/B be an FLinvariant section of G such that A > C > B defines a fixed F-composition series of A/B. If property (C) holds for this series, then property (C) holds for every F-composition series of A/B. Proof. We may assume that a second F-composition series of A/B exists and is defined by A> D > B where D I: c. Then we must have CD > C > C n D, so the fact that C and D are F-invariant shows that A = CD, and B = C n D. (26) Therefore A/B ::: C/B x D/B, A/C ::: D/B, A/D -:::. C/B, where the decomposition is an operator decomposition, and the 1somorphisms are operator isomorphisms. Suppose the decomposition (26) is unique. I f F/CF(A/D) lies in 1p' it follows from (26) that F/CF(C/B) lies in "J.P. Since property -21- (C) holds for the series A> C > B, F covers C/B. (F n A)D ~ (F n C)D ~ CD = A, so F covers A/D. Therefore Suppose F/CF(D/B) lies in 1.q , then F/CF(A/C) lies in 1.q • By property (C), F covers A/C. Because of the uniqueness of the decomposition, and the fact that F n A is not contained in c, either A = (F n A)B, or D = (F n A)B. In the former case, F covers all of A/B, hence F covers the subgroup In the latter case, F n A = F n D since F n A ~ D, so D/B. D = (F n D)B. Therefore, in either case, F covers D/B. The decomposition (26) is unique if CIA/cj,jc/Bj) = 1, so we may assume A/B is an elementary abelian p-group for some prime p. This means that we can look at A/B as a Zp (F)-module. If A/C and C/B are distinct Zp (F)-modules, then it follows from (13) that the decomposition is unique. Thus we may assume that D/B ~ z · (F)9/B .• . p . In this case, it f'ollows from property (C) f'or A> C > B that F either covers, or avoids A/B. Lennna 4. Therefore Lemma 3 holds in all cases. Assume F E 1, H:: G, and F ~ NG(H). If' property ( C) holds for a fixed F-composition series of H, then it holds for every F-com- • position series of H. Proof. Let H = H0 > H1 > • • • > Hn = 1 be the f'ixed F-composition series of H for which property (C) holds. We use induction on n. The lemma is. true if' n = 1. Let H = K0 > Ki > .~. > Kn= 1 be a second F-composition series f'or H. If' Ki = H1, property ( C) holds for the series -22- K0 > Ki > • • • > Kn = i by induction. Suppose Ki # Hi, and let i be the smallest integer such that K1 n Hi = Hi. Because Hi ~ K1 n Hi-l < Hi-l' we have Hi = IC1 n H1 _1 , so that we have the f o11owing lattice diagram; H i Since H1 is part of an F-composition series for H, F :S NG(H1 ). Be- cause of the isomorphisms indicated in the diagram, Hi > Ki n Hi > • • • > K1 n Hi-l = Hi > Hi+i > • • • > Hn F-composition series for Hi which has ·length n-1. property (C) holds for this series. = 1 is an By induction, Therefore, property (C) holds for the F-composition series of H/Hi n K1 defined by the series H > Hi > H1 n K1 • By Lemma J, property (C) holds for the F-composi- tion series H > K1 > 'i n Hi > • •• > K1 n Hi-l = Hi > ••• > Hn of H. =1 In particular, property (C) holds, by induction, for any F-composition series of K • Therefore property (C) holds for ~he 1 series K0 > K1 > • • • > Kn = 1. -23l'heorem 4. · I.et 1 be a formation locally defined by f1 ). p a group, F $. G, and F EC!f. Let G be If property (C) holds for a fixed F-composition series G = G > G1 > ••• > G = 1 of G, then Fis 0 n an 1-subgroup of G. Proof. We use induction on !GI. By Lemma 4, we may assume that the series G = G0 > G1 > ••• > Gn = 1 is a refinement of the chief series G = ffo > H1 > • • • > Hm-l > Hm = 1. k. Then Hm-l = Gk for some l\n-l is a minimal normal subgroup of G, so we set Gi =Gi/Gk = o,1,•••,k, F = FGk/Gk, and G =Go· Our first step is to show that F is an 1-subgroup of G. If m = 1, then -G = F =-1, so the resul.t is trivial. I f m > 1, for i then Hm- 1 < G, and G = G0 > G1 > • • • > Gk = I is an F-composition series of G. By induction, to show that F is an 1-subgroup of G, it is sufficient to verify property (C) for the series -G0 > G1 > • • • > -Gk =-1. For each i, set Ki= Gi/Gi+l' and Ki= Gi/Gi+l• for i < k, Gk centralizes the section K1 for i < k. C,(Ki) = CF(~)Gk/Gk for all i < k. Thus, F/CF(i1 ) =:: FGk/cF(Ki)Gk::: F/CF(K1 )(F n Gk). But F n Gk ·s CF(K1 ), so we have (27) F/c,<ii) =:: F/CF(Ki) for i < k. Since Gk$. Gi+l Therefore, -24- Suppose Ki is a p-section of G such that F/°f(K1 ) E "JP • (27), F/CF(Ki) E "JP , so F covers Ki • By Therefore, (FGk n Gi)Gi+1 = (F n Gi)GkGi+1 = (F n Gi)Gi+1 = Gi • By taking homomorphic images, and noting that FGk n Gi/Gk = F n Gi, we get - - Thus F covers Ki • Therefore property (C) holds for the F-oomposition series G = G > ••• >Gk= I of G. 0 Now that we know Fis an°1-subgroup of G, it follows from {2.5o) that we can complete our proof by showing that F is an 1-subgroup of FGk • Suppose FGk < G. where Di We consider the series FGk = D0 ~ D1 ~ • •• ~Dn =1, = FGk n G1 for each i. Suppose Di > Di+i tor some i. Then This is an operator isomorphism, hence because F is irreducible on Ki' we have {28) Therefore the distinct terms of the series, D0 ~ D1 ~ ••• ;::. Dn = 1, form an F-composition series for FGk which passes through Gk. Since F covers FGk/Gk' and since Di = Gi for 1 ~ k, property (C) holds for this composition series. Suppose .G By induction, F is an 1-subgroup of =FGk. Since Gk is a minimal normal subgroup of G, F acts irreducibly on Gk. Therefore F either covers, or avoids Gk. -25I f F covers Gk' then F = G, so F is an 1-subgroup of FGk. avoids Gk' and Gk(=Gn_ 1 ) is a p-group. Suppose F Then FGk/CFG (Gk~ F/CF(Gk) k cannot lie in "fp since property (C) holds for Gk. Therefore Since FGk/Gk lies in 'f-, (FGk)'} = Gk. (FGk)'f 2::, Gk. I f F :SU :S FGk' then U = F, or ·u show that F covers U/U1 in both cases. = FGk. The above remarks Therefore F is an 1-subgroup of FGk' and the proof is complete. As one application of Theorem 4, we prove Theorem 5. Let 1 be a formation locally defined by £1 ), H E 7, and p let I be a finitely generated Zp (H)-module. Let G = HI be the semi-direct product of I by H where the action of H on I by conjugation is the canonical action given in Chapter II. (a) F = HC1 (o ,(H )) is an 1-subgroup of G. (b) As a Z (H)-module, I= CI(O ,(H 1p p p Proof. Then, p Let W = c (o ,(Hn )). 1 p Tp 1p )) 19 G. Our first task is to show HW lies in 1. Suppose q # p, then W is a q•- group, so 0q , ,q(F) =0q , ,q(H)W. Therefore, F/Oq,,q(F) Qt H/Oq,,q(H). Since H lies in 1, F/oq,,q(F) E 1-q. Let U = H • Then 0 , (U) centralizes W. Since H/O , (H) lies 7p p p ,p in p U < 0 ·, (U). Therefore 0p , ,p (UW) = UW. UW is norma1 in F, - p ,p 1, so by (18'), UW -< 0p , ,p(F). Therefore F/O p , ,p (F) is isomorphic to -26a quotient group of H/U. Since H/U E 1_p , F/Op , ,p(F) E 1p . Therefore, F lies in "f. Now let G = G0 > G1 > ••• > Gn for G such that G~ =I for some J. = 1 be an F-composition series In order to check property (C), we need only consider Ki= Gi/Gi+l for i ?;.}., since F covers G/I. W centralizes every K , so we have 1 If i?::. J., and F/CF(K1 ) E 1p' then CH(Ki) ~ U. Op 1 (U) centralizes Ki. (17), W covers Ki. In particular, I is a ,p -group, so by the covering lemma, Therefore F covers Ki' so property (C) holds for the series G0 > G1 > ••• > Gn = 1. By Theorem 4, F is an 1-subgroup of G. We shall prove the next assertion by induction on the length of a Zp(H)-composition series of I. Since H =:: G/I, G1 s; I. Since F covers G/G1, it follows that, as a Zp (H)-module, I = W+ G1 , so. the major difficulty is to show that the sum is direct. If I is irreducible, one of the two summands must be the zero subspace, so the theorem is true i f n = 1. I Suppose I = I 0 ::> I 1 :::> • • • ::::>In = (O) is a ZP( H)-composition series for.I, and n > 1. n- 1 is a minima1 normal subgroup of G, and Hin_ 1/In-l acts on the quoti~nt module I = I/In-l in the same wa:y that H acts on I. I Therefore G = G/In-l is isomorphic to the -27semi-direct product HI where the action or H by conjugation is the module action. By induction, i = Cy(Op' (U)) ~ (G)"J • (29) cASE r. (o> c eycoP, cu» s r. Set T/In-l = (G)1• By the covering lemna, C...(O ,(U)) =WIn- 1/In- 1 • -~ p contained in I By (29), T (17), n Wis a norma1 subgroup of G • n- 1 I.f T n W = (0), then I = T ~ W, and reverting to multiplicative notation, we have F n T = 1. Therefore, T is clearl:y the smallest normal subgroup of G whose factor group is covered by F. Since F is an 1-subgroup or G, it follows that T = G1• Suppose T n W = In-i• The hypothesis or CASE I states that T c I, so the theorem is true, by induction, for the semi-direct product G1 = Hr. Therefore, T = CT(OP (U)) E9 T1 , where T1 = (G1 liq. 1 Now In-i = CT(OP 1 (U)), and because I is abelian, T1 <l G. Therefore, as a zp(H)-module, AB before, this forces T = G1 since G/T Cl Fis the largest possible 1 1 factor of G covered by F. Therefore, the theorem holds in CASE I, By assumption, Ws; In_ • 1 OP 1 (U) is normal in H, so Wis normal -28- If' W = (0), then His an"'l-subgroup of G, and I a G • in G. 1 may, therefore, assume that W = I ,. Let I be the dual of I. ,. n- 1 We • By (8) ,. l l /."'- In-1 ~ I/In-1 ' and In-1 ~ (I/In-1) Since Op,(U) centralizes In-l' it follows from (9), that Op 9 (U) ,. centralizes I 1• By the covering lemma, (17), CI"(O ,(U)) covers nP I/In~1 • Now Op 1 (U) ~ H, so CI l (0 ,(U)} is a Z (H)-submodule of I 11 • p ~1 p ~ By (8)' I/[CI 1 (0 ,(U))]l n-1 P ~ [~]. . n-1 P Since 0 ,(U) centralizes CI 1 (0 ,(U)}, it follows from (9), that P · n-1 P · 0 ,(U) centralizes I/[CI l (0 ,(U})Jl • P n-1 P Thus using the covering lemma. again, we lmow that W covers the factor I/[CI 1 (0 ,(U)}]l • n-1 But CI l (OP, ( U)} S In~1' so W = In-l £; [CI l n-1 . n-1 p (0 , ( U)} ]1 , hence P we must have I :: [CI 1 (0 ,(U))]l • n-1 P Therefore CI l (0 , (u)) :r (O}, so C"'I(O , (U)) avoids I 1 1 • These nn-1 P P statements show that -29- But by (10), this splitting induces a splitting in I, I = [cI~(op ,(U))]l EJ) I n- 1 it follows from the fact that F = HI is an n- 1 1'- subgroup of G, that G1 = [CI(Op,(U))~ • This completes the As before, proof in ail cases. Remark. This result cannot be extended to the case where I is a p-group of class 2 because of the following example. quaternion group. Let I be the I has an automorphism, h, of order ) such that h acts fixed-point free on I/D(I), and centralizes D(I). Let H be the cyclic group of order ) generated by h, and let G = HI. A Carter subgroup of G is H x D(I), but D(I) has no complement in I, so no splitting is possible. -30- Chapter IV Strong Containment In this chapter, we shall characterize strong containment. In certain cases, we can make our characterization more precise by giving generating sets for certain of the formations 1. in the minp imal local definition of "J. The results of this chapter form the basis for our results in Chapter V. Let Lemma 5. e and 1- be two non-empty saturated formations; let. e be locally defined by [ epJ. Let G be a group of minimal order satisfying An e-subgroup of G is not contained in any1-subgroup (P) of G. If F is an '1-subgroup of' G, and E is an e-subgroup of F, then (a) G1 = M is a minimal normal subgroup of G; G is the semi-direct product of M by F; F acts faithf'ully and irreducibly on M. (b) If Mis a p-group, then E* = ~(op,(E e. P)) is an £-subgroup of G, and 1 < CM(Op,(Eep)) ~ M. Proof. If G E "/, then G = F contains every t-subgroup of G, hence G does not satisfy (P). the identity. Therefore G ~ ~; in particular, G is not Let M f. 1 be a minimal normal subgroup of G. (2.5b), FM/Mis an 1-subgroup of G/M. By Because of the minimality of IGI with respect to the property (P), some e-subgroup of G/M is -31- contained in FM./M. Since all £-subgroups of G/M are conjugate, we can find an ~-subgroup, E, of G such that 'EM· ~ FM. f-subgroup of G, is also an t-subgroup of FM. E, as an Because G satisfies (P), no conjugate of E under FM can be contained in F. Therefore FM~ G, and satisfies (P), so G =FM. G/M lies in 1, but G does not lie in 1, so G "J. = M. Since F fl M is a normal subgroup of G, properly contained in M, F n M = 1, so G is the semi-direct product of Mby F. Since Mwas arbitrary to begin with, and we show9d M = G"l, M is the unique minimal normal subgroup of G. Therefore F acts faithfully and irreducibly on M. This proves (a). G/M is isomorphic to F, so FJII./M is an C-subgroup of G/M. (25c), an 2-subgroup of FJII. is also an e-subgroup of G. By By Theorem 5, E* = E~(Op,(Ee )) .is an e-subgroup of EM. Since E* is not contained p in F, 1 < ~(op,(E£ )) ~ M. p Before stating the characterization, we introduce some notation. Definition 6. If e and 1 are two saturated formations, and eis locally defined by ( ep) , set (a) of e. (b) n(l) = (pjep is non-empty}. n(e) is called the characteristic I f p E n(l), set -32- tp H has a faith£ul. irreducible Zp (H)-module, M, such that if E is an e-subgroup of H, then = 1 < ~(OP,(E~)) SM. (c) If p E n(l), set 0 =jH E t P · There is at le•st one faithful irreducible ZP(H)-module, p M, such that 1 < ~(O ,(Ee )) < M. p Theorem 6. p Suppose ! and 1 are two saturated formations locally e ) and { 1p ) respectively. Then defined by { p ~ << 1 <=> t pc- 1p for each p E n(e). Proof. Suppose tp S u.f.p for each p E n(e), and contained in e is not strongly 1. Then the class of groups satisfying property (P) is non-empty, so w~ choose G to be an element of minimal order in this class. By Lemma 5, if GU/ is a p-group, then p divides the order of an e-subgroup of G, hence p E n<e). is an 1-subgroup of G, then F lies in t p • By Lemma 5, if F Therefore, F is an element of 'J.p • Since G'J is the unique minimal normal subgroup of G, G~1 = 0 p , ,p (G). There:f'ore F z G/Gnt = G/O p 1 ,p(G) lies in 1.p • If' q # p, then G~ < 0q ,(G), so 0q 1 ,q (G) = G~Oq , ,q (F). . 1- Therefore, G/Oq 1 ,q (G) z F/Oq 1 ,q(F) Since FE"}, it follows that G/Oq , ,q (G) E 1. q By (O''), G lies in -33- 1, a contradiction to the £act that G'1 > 1. Therefore e<< 1. Suppose ~<< 1, p E n(e}, and F ~ tp. I.et M be the faithful. irreducible Zp (F)-module mentioned in the definition of t p • G Set = FM, where the action 0£ F on M by conjugation is the module action. By Theorem 5, an1-subgroup of G is F* = F~(O ,(F1. - )), p hence G = F"'M. I.et E bll an e- subgroup 0£ F. Since EM/M is p an e-subgroup 0£ G/M, it follows from (25c) and Theorem 5, that E* = E~(O ,(Ee )) is an e-subgroup of G. E* does not avoid M, p and because p r << "J, E* is contained in some '/-subgroup of G, hence F* does not avoid M. Since 0 ,(F~ ) ~ F, ~(O ,(Fn )) is normal p in G, hence F• =G. p tp 3p Since G is an element of 'J,and F acts faithtul1y on the p-chief factor, M, of G, we have F:::: G/CG(M) E "JP. tp s: 1p· Because of this characterization, i f Therefore e<< 1, and p E n(e), then tp £ 1p for a:ny "Ip which lies in some local definition. leads naturally to the question: local. definition for This Suppose ("/PJ is the unique minimal 1. I f p E n(e), is 1P the smallest formation generated by the set t p ? The answer to this question is yes, pro- vided e p is non-empty for at least two primes. able to relax the hypothesis on the ep•s. We have not been In the next few 1emmas, we shal1 investigate properties or the tp's and ep•s as a prelude to giving a partial answer to this question. -34Lemma 6. Let e and 1 be non-empty saturated format:ia:ls with local definitions {er) and {11p) respectively. Suppose e<< 1, and is G an element of 1 withe-subgroup, E. (a) Suppose GE ~ • (b) I f q E n(e), Oq(G) = 1, and the pennutation representation, 1 (E 0 q• q tq G lies in 0 q if, and only if, 0 ,(E0 ) > 1. q ) IG, is faithful, then G lies in ~q (t q ) , the smallest for- mation generated by the set t q • (c) Let M be a faithful irreducible Zp (H)-module. 10 (E q' (d) If G = HM, and eq )IG is faithful for some q in n(l) - {p), then GE t q • For each r,s in n(e), er s (ts). Proof. Suppose G E t • q Then G has a faithful irreducible Z (G)-modq ule, I, such that 1 < CI(oq,(Ee )) s I. I f J is any such module, q J = CJ(oq ,(E~C.q )) if; and only if, oq ,(Eeq ) = 1 since J is faithful.. This proves (a). Let G be a group satisfying the hypothesis of (b). Let T be the z (G)-module which gives the representation 10 (E )IG. q q• t By (15), q we can write T as a direct sum of principal indecomposable Zq (G)-mcrltiln, For each k = 1,•••,s, it follows from (16) that Tk has a unique maximal proper Zq(G)-submodule, ~ • We set Jk = Tk/.r:r;, • -'.35Since 0q (G) = 1, F(G) is a normal q'-subgroup of G, hence by Theorem 1, the distinct irreducible components of JklF(G) are exactly the same as the distinct irreducible components of TklF(G)• Therefore, since Tis faithful, it follows that if J = J 1 ~ ••• $ Js' By Lemma 1, J is a faithful Zq(G)-module. then JjF(G) is faithful. We now apply the Frobenius recipr?city theorem for modules. For each k = 1,•••,s, (0) c Holllz (G)(T,Jk) :: Holllz (O (E ))(1,'klo . (E )) q q q' ~ q• e q q Therefore 1 < CJ (0 ,(E0 )) ~ Jk' for each k. k Gk. q Cq Set Gk= G/CG(Jk). Then ~ = ECG(Jk)/CG(Jk) is an e-subgroup of By Lemma 2, (~)~ = Ee cG(Jk)/cG(Jk). q Therefore oq,«Eic>e ) • q q Oq ,(Etq )CG(Jk)/CG(Jk), so . 1 <CJ (0 ,(Ee )) = CJ (0 ,((~)e )) ~ Jk. k q q k q q Since G lies in "f., q E n(e ), and Jk is a faithful irreducible Zq(Gk)-module, Gk lies in 4>q for each k. lies in (t~ Therefore G = G/Q CG(Jk) , the smallest .formation generated by the set tq• This proves (b). · The proof of (c) is essentially the same as the proof of (b). \ Let G = HM be the group mentioned in the hypothesis of {c). Let T be the Z (G)-module which gives the faithful. representation 10 q . . . . q' (E ; )I- G• -36Once again, T has a decomposition into a direct sum, T = T EP ••• E& T , 1 8 of principal indecomposable Zq (G)-modu1es. . Because of faithfulness it follows that Mis non-trivial on some Tk' say T1 • minimal normal in G, M must act faithfUlly on T1• Since Mis Since M is a q•-group, it follows from Theorem 1, that if J = T1/JT";,_ , then JIM is faithful. ~t Mis the unique minimal normal subgroup of G, and by (14), 0p (H) .= 1, so F(G) irreducible Z (G)-modu1e. q = M. By Lemma 1, J is a faithful By Frobenius reciprocity, 1 < cJcoq,<Ee q »s J. Therefore, the fact that q E n(e) says that G is, by definition, an element of ~ • q This proves (c). Part (d) is the only statement in Lemma 6 which requires the assumption e<< 1. Suppose H E er' E is an e-subgroup of H, and M is a fai thfu1 irreducible Z (H)-modu1e such that 1 < CM( 0 , (E 0 ) ) < M. r r ir Set G = HM. By Theorem 5, F = H~(Or' (~ ) ) is an 1-subgroup of G, r . and since E:::: FYJ./M is an e-subgroup of G/M, E"' = EX:M(Or,(Ee )) is . an e-subgroup of G. is an element of G. Since r e<< 1, F cannot avoid M, hence F = G 1. Let N be the intersection 0£ al.l. the conjugates of os,(E"'e ) in . s Then N ~ G, and N n MS E"' n .M = ~(Or,(E~r)) < M. Therefore N n M = 1. This shows that 10 s' (E* e8 ,1 GIM is faithful.. . Because I -37- M= F(G), it follows from Lemma 1 that 10 (E• )IG is faithful. s' 2, s By part ( c), G f t 5 for s E n(e ) - {r). element of (ts) , for s E n(e)- { r). that er S (ts) for each s in n(e). Therefore H :::: G/M is an Since er S tr' it follows This proves (d). The next lemma has an elegant proof. This proof was told to me by Dr. E. C. Dade, and it shortens this part of' the original proof of the next theorem considerably. Lemma z. Let A, B be two groups and assume Z(A) = 1. If M is a faithful Zp (A)-module, and T is a faithf'ul. Zp (B)-module, then M@ T is a faithf'ul. Z (A x B)-module. p Proof. I f V is a vector space over ZP, we let GL(V) denote the general linear group on V. Then A x B $_ GL(M) x GL(T) = C, so we examine the kernel, K, of the representation of Con M® T. Let m1 ,•••,mr be a Zp -basis for M, and t 1 ,•••,ts be a Zp -basis for T. Then{~© tjl1 $. i $. r, 1 $. j $. s} is a Zp-basis for M®T. Suppose f x g E K, and for each i tjg = l YjitR; Then m1 ® tj = m1 ® tj(f' x g) = for each j. (l cpiklllic) ® (l y jRt.e ). So -)8- mi® tj = l cpiky jJ ~ t..f, hence we have ® k,~ o if (i,j) # (k,R) 1 if (i,j) = (k,~) cpiky jR = Therefore m1 f = cpmi for each i, and tjg = cp-1tj for each j. Suppose a x b E (A x B) n K. Since Z(A) = 1, and our results show that K ~ Z(GL(M)) x Z(GL(T)), we must have a = 1. means that the constant, cp, is the identity, so b But this = 1. Therefore Ax B acts faithfu.lly on M® T. Lemma . 8, Suppose A E ~ p - 0p , B E 0p , and either Z(A), or Z(B) is the identity. Then A x B E (ep) , the sma11est formation generated by the set e • p Proof. Let E be an C-subgroup of A, and E an C-subgroup of B. AE ~ - 0 , it follows from Lemma 6a that O ,(Eo ) = 1. p p p in 2, so E/O , p ,p p p - p Since(E x"NJ/(¥.0 subgroup of Ex 'I contained in C.p x ~n) E c;p p p . We wish to examine g, (Ex ~)e is a · normal. p p Ee x 'le. we ·def'ine a subgroup, W, p of E'ep bys p ;:P' Since B E e , 0 , (l!'E. ) > 1. Now E x }' is an e-subgroup of A x B. 0 ,((Ex mo). p C.p But E lies (E) is an element of 8 , consequently Ee, < O , . (E). Therefore E () is a p-group. tp c.p Since p W = {e E E'ep13t E (EX }')ep ~1 t = dxe, . and d E Elp}. -39In other words, W is just the collection of al1 elements of IE. p which appear as components of elements of (E x I)£ • W is clearly p a subgroup of le , and since (Ex l)g is normal in Ex~' Wis p normal in 1. p By construction, (E x l>e ~ E x w, hence p l/w ~(E it)/ (E x w) lies in ep. Therefore w = I£ • p Now if e is any element of 0 ,(It,), then there is an element p p . Ee din E~, such that t = dxe lies in (Ex ~>e. Ent is a p p p p-group, hence by taking an appropriate power of t, we see that e lies in ( E x ~ e • Therefore, p <JO> o ,<'Ell >!S: o ,C<E x l>e >!S: o ,(En x le >=.o ,Cl~ >• P c.P P P P cp P P P By assumption, A has a .faithful irreducible Zp (A)-module, M, . and B has a faithful irreducible Zp (B)-module, T, such that 1 < CT(O ,(lf't )) < T. p tp By Lemma 7, M ® T is a faithful Z (Ax B)-module. p Since M ® TIB is isomorphic to a multiple o.f T, i f we let U be any Zp(A x B)-composition factor of M® T, then UjB is also a multiple of T, so because of (JO) we have (31) 1 < Cu(Op ,Cln(p)) = Cu(Op ,((E x ~n(p)) < u• . for each u. · -40- Let G =(A xB)/CAxB(U), then E =(Ex l)cAxB(U)/CAxB(U) is an ~-subgroup of G. By Lemma 2, iep = (Ex 'r.)ePCAxB(U)/CAxB(U), and by (31), O ,((Ex E)e ) does not centralize u, hence O ,(Ee)> 1. p p p p Since 1 < c0 (o ,((Ex l)e )), and G is an element of "f., it follows p p from Lemma 6a that G lies in ep • Let V be the direct sum of all Zp (A x B)-composition factors occurring in a compa;iti.on series of M®T. :av- (14), F(A x B) = F(A) x F(B) is a p'-group, so the fact that M ® T is faithful implies VI F(A X B) is faithful. By Lemma 1, V is a faithful completely reducible Therefore(A xB)/~ CAxB(U) lies in (ep) , where ZP(A x B)-module. the intersection is taken over the irreducible co~ponents of V. Since V is faithful this intersection is the identity, hence A x B E (ep) • Corollary 1. If e<< 1, and there is an element B in 0p such that Z(B) = 1, then (tp) s (t~for each q in n(e) • . Proof. By Lemma 8, if A E tp - ep, then A x B E (ep) • Therefore AE(eP) , so (tp) = (eP) . By Lemma 6d., if q ETT(e), then ep S (~q) , hence (tp) Theorem 7. . Suppose S (tq) • e<< 1, and ep'. 0r are .non-empty for two primes p,r in the characteristic of definition of 1. Then e. Let (1. } be the unique q . minima] local -41- for each q in ncl). Proof. We define a new forma.tion,1 , locally by setting (32) for q E n(E)• e<< J, f}q S 1q for each q, by Theorem 6. Therefore~ S 1• Since Let i s be the set specified in Definition 6 for the formation Since 'j S ~ , is S ~ s S ff. e<< P/, by Theorem 6. Suppose 1s for .each s in n(e). Therefore 1c1. If G is an element of minimal order in 'J. - ?, then G is a semi-direct product, G = 1l'M, where 'i1 is an '7-subgroup of G, and acts faithfully and irreducibly on the elementary abelian t-group M. Since .G lies in 1- 7, 'i1:::: G/CG(M) lies in 'Jt - 7t • Fort in n(e)• this contradicts the definition of ~t' hence t E ncE). Since e<< tJ, 1:", as an ~-subgroup of G, must contain some e-subgroup, E, ~f G. tation 1 0 . q' q in n(r) - Thus for any prime q, the permutat:-on represen- (E ) IG is faithfu.1. eq By Lemm.a 6c, G lies in tq for each {t}. By Lemma. 6d, eq s'Js for each q,s in n(e), hence i f GE eq for some q in n(r) - (t), GE 1't• for each q in n(C) - {t}. say p. Suppose, therefore, that GE ~q - eq One of the primes p,r is unequal tot, Then G is an element of ~P - ep such that Z(G) ~. 1. Since -42- ep is non-empty there is a group H in 9 p , so by lemma 8, G x H E /9 \'p ) • By Lemma 6d, G x H E 7t' hence in each case 'r, as a factor group of G, must lie in 'f.t' a contra.diction. Therefore 'f =1. Since {'jq }forms a local definition for 1, we have ~q £ 1q £PJq for q En(£), so the proof of Theorem 7 is complete. Because we could not relax the hypothesis on the e 's, we thought p it appropriate to include e<< 1 , and p E n(t). ep is empty if, and only if, for each element F of 1, an e-subgroup, E, of F either covers, Theorem 8. Suppose or avoids each p-chief factor of F. Proof. Suppose an f-subgroup of F either covers p-chief factor of F .for every Fin or avoids each 1. Let FE tp' and let Ebe an ~-subgroup of F • . Let M be a faithful irreducible Zp(F)-module such that ~(Op 1 (E;)) > 1. . F* = F~(Op ,(F 1p 111 )) does not avoid M. By Theorem 5, and the fact that e<<1, is an 1-subgroup of FM, acts irreducibly on M, and · Therefore F* =FM; M is which is not avoided by thee-subgroup, E* a P-chief factor of G= FM = E<1i(Op,(E~)), of G. = ~(Op ,(EaCp)),. so e is empty. . p Suppose ep is empty, F E 'J, and E is an e-subgroup of F which Therefore M does not avoid the p-chief factor K = L/N of F. Let F = F/CF(K). Our first assertion is that the semi-direct product 'FK lie.s in. -43- 1 (the action of F on K is the action induced by the action of F on K). By Theorem 5, F* = FCK(O ,(F..,,. )) is an 1-subgroup of FK. ThereP -t.P. . fore 'F* acts irreducibly on K, and F*/CF*(K) ::::'F. · Since FE 1, F E '1p by definition. By Theorem ) , F* covers K, hence FK lies in 1. E = ECF(K)/CF(K) is an e-subgroup of F. By Lemma 2, Ee = Ee cF(K)/cF(K). p p 0 p Because o , p ,p (El1 ) = Ee , it follows that p p ,<'Erp ) = 0p ,(Eep )CF(K)/CF(K). Therefor~ CK(Op ,(Eep )) = CK(Op ,(El'p )). O ,(Ee ) centralizes every p-section of E, hence O ,(Ee ) centralizes p p p (L n E)N/N, a non-identity subgroup of K. p Therefore, 1 < CK( op, (E~)) !S, K Thus F lies in ~ p • ep is empty' so it follows from Lemma.- 6 that Ee is a p-grou~. If U is any E-composition factor of K, then - p Eo centralizes U·since 'p Erp ~ 0p (E). Upon taking inverse images in - E, we see that CE(U) 2:, Er , so that E/CE(U) lies in ep• . p E covers U, hence E also covers all of K. . By Theorem J, -44Chapter V Structure Theorems Throughout this chapter we shall make the following assumptions: HYpothesis I. eand 1 are saturated f Grmations such that (a) "il s e<<1; (b) there is a non-empty formation,~' such that Our first theorem saJlS that the structure of e= fG E _g IG/F(G) tf}. 1 is essentially the same a.s the structure of E in that we can show the existence of a formation, 'IJ , such that 1 = ( G E J IG/F( G) E lJ}. We first prove two lemmas. Le™ 9. I.et °I be a non-et11pty formation. loca.lJ.sr defined by setting JJ Then p e=JJ. Proof. Suppose GE e. I.et~ be the formation =1 for each p. I.et E= (G. EJ IG/F( G) E1J. Because 0 , (G) 2:. F(G), G/F(G) E ~implies p ,p . that, for e'a ch p, G/Op , ,p(G) lies in 'f. By (cr'), G lies in JJ. 'f: Since 'f is a formation, and F(G) = n O , (G), G/F(G) lies inf. Therefore p p ,p Suppose G E J,J. a E e, so e = JJ • Lemma. 10. Suppose (a) Then, for each p, G/Op , ,p (G) E e, 'J. satisfy Hypothesis I. I f G is a group with f-subgroup, E, and E E1, then E = G. -45- 1, and G is an element of T such that 0 (G) = 1, then G lies in 1. q q (b) If f1q} is any local definition for Proof. We prove (a) by induction on the nilpotent length 0£ G. G is nilpotent, there is nothing to prove. EF(G) lies in!. Since E lies in 'f, Since Eis an e-subgroup of G, E covers U/U~ for any subgroup, u; of G which contains E. Therefore E > F(G). G = G/F(G)' then E = E/F(G) is an e-subgroup of 'G. E I:r Set By induction, = G, hence E = G. Let f1 } be any local definition for 1. q Op(G) = 1. Let M be the regular Zp(G)-module, and form G = GM. 1 Since G lies in 'i, G1 lies in strong containment that G ~ G /M, G lies in 1 e• It is a simple consequence of eS 1, hence G1 E 1. Since Op(G) = 1, .and G acts faithfully on M, M Theorem 9. Suppose G E '(, and = 0p , ,p(G1 ). Therefore G /M E 1 1p . Since 1p• This proves (b). Suppose f and 1 satisfy Hypothesis I. f orma.tion, 1J , c~ntaining °1"',such that Then there is a 1= f G E J IG/F( G) E V}. t =1, the formation 'Y satisfies the requirements of the theorem. Assume e c 1. By Lemma 9, we know that the .family, re p Irp =. "r for each p}, of formations is a local definition :ror t Proof. If We shall use this family for the local definition of the remainder of the proof. definition of Let ethroughout £1q } be the wlique minimal local 1. A second application of Lemma 9 says that we need -46- 1 only show 1. = r s for each pair of primes, r,s. In view of Theorem 7 and Corollary 1, we begin by examining the set e for various s primes s. Since'1.se , n(e) contains a11 primes, so e and~ s s are defined for each s. Let G be an element of minimal order in ct/- t. By minimaBty, if N is any norma1 non-identity subgroup of G, then G/N lies in Therefore Ge is the unique minimal normal non-identity subgroup If E is an e-subgroup of G, then Ene of G. e. = G, and En Ge = 1. Furthermore, E acts faithfully and irreducibly on Ge • We set M = Ge_,, and note that M is an elementary abelian p-group for some ·prime p. Since G is not in e, E cannot lie in 'f. Therefore F(E) ~ E'r> 1. But it follows from (14) that F(E) is a p'-group, so for some prime r I: p, E'l' has a no~-identity normal Sylow r~subgroup, R~ I f s '/: r, then Because M is the unique minimal normal subgroup of G, and E n M = 1, 10 s' (E es )IG is faithful for each s. By Lemma 6, G lies 'in e for 6 each prime s I= r,p. Since E is .faithfu1 and irreducib1e on M, Z(G) = 1. We now choose a prime s ~ r,p. Then G E es, so there exists a faithful irreducible Z6 (G)-module, J, such that 1 < CJ(0 8 ,(Et )) < J. 8 -47Set G1 = GJ. Since Eis isomorphic to an e-subgroup ~f 'G 1 /J, it follows from (25c) and Theorem 5 that E* = ECJ(Os 1 (Ee )) ·is an . e-subgroup of G1 • s 1; An ~-subgroup of G it cannot avoid J because f!,<< 1. covers G /J since GE 1 1 Therefore G1 lies in 1. Because Eis a quotient group of E* which does not lie in 'f, E* cannot lie in "r, hence 1 < (E*~= (E*>e ~ Ei.rcJ(o9 ,(Ee )). s p (E*):y is a p'-group because ~is a subgroup of the p'-group, F(E), . and s '/: p. 1(E*~I G1 is faithful since J is the unique minimal normal subgroup of G1, and (E*)"l'n J ~ CJ(o9 ,(Ets)) < J. from parts {a) and (c) of Lemma 6 that G1 E ep. It follows By construction, Z(G1 ) = 1, hence we have established (33) I f s i= r, t~en there is a group, X, in es such that Z(X) = 1. We now apply the results of Chapter r:v. n(e) contains all primes, so it follows from Theorem 7, and (33) .that each prime· s. 1s = (t 5 ) for By Corollary 1, we have for s,q in r' (34) 1 ts -c: r Qf For s i= r, we set 1 = 'lJ. s show ~ r c:'\}. £or each s. The final step in the proof will be to By part (d) of Lemma 6, er S 1s for each s, so er s;1J. Suppose -48- HE t r - er , and E is an e-subgroup of H. E ~ = E"r is an r-group. = H, and i f s is any prime not dividing rl HI, 0 (H) = 1, so H E 1. = iJ. s s Suppose E'I = 1, then E lies in <Ii'l • Suppose Er> 1. By Lemma 10, E By (14), Or(H) = 1, so 1&,-IH is faithful. Since H E 'r' H also lies in 1. Thus if s is a prime which do.e s not divide I HI , it follows from part (b) of Lemma 6 that H E (ts) • Therefore tr s'V ~1r =(tr). Since 1J= 1s for each s, Lemma 9 says that 1-= {G Ei IG/F(G) EV). The fact that V contains 1 is a consequence of part (b) of Lemma 10. We are interested in finding formations which are maximal in the partial ordering <<. assume Since e<< 1 implies l S 1, we shall . ec 1, as ~ell as Hypothesis I. Since e::: {G Ei I G/F(G) E 'TJ' we shall fix our local definition fore by setting ' e =1 for each p 1,p ) is the minimal local definition for 1. By the proof of Theorem 9, there is a formation, lJ, containing 'f, such that 1 = \) for each p. Since ec 1, we must have 1 c 1J. p p. We shall assume that { Before stating our main .theorem, we prove several lemmas. The proof· of Lemma 12 contains the essential. constructions used in the proof of our main theorem. Lemma 11. Let G be a group, and 1 < H ~ G. Assume that the permu- -49tation representation, 1HIG, is faithful. If Mis the Zp(G)-module which gives 1HjG, set U = n ~(H)g. Then U is a Zp(G)-submodule gEG of M, and M/U is a faithful ZP(G)-modu1e. Proof. We can choose the cosets of H in G to be a Zp -basis for M, i.e., M = Zp ·H .~ Zp .Hg2 ~ ••• ~ Zp .Hg s , where s = [G:H], and the oper- ation of G on M is by right multiplication. For each g in G, CM(H)g = ~(~), hence U = n ~(~) = ~( U ~). gEG In other words, if N is the normal closure of Hin G, then U gEG = ~(N). Since N < G, U is a Zp (G)-submodule of M. For i = 1,•••,m, letLJi denote one of the orbits of the cosets H, Hg ,,••,Hgs under a~tion by N. Since N < G, G permutes these 2 orbits transitively, thus all orbits have the same number of elements. The number of elements in a:ny orbit is [N:H], and since 1HIG is faithful, H cannot be normal in N, so [N:H] ~ J. set LJ1 = (~ 1 ,•••,Hg1r} where r For each i, we = [N:H]. r Let u1 = l Hgik' for i = 1,···,m. Our first assertion is that k::1 u , ••• ,um form a Zp-basis for c.._(N). The elements u 1 ,•••,um are· -M 1 clearly linearly independent, and since ()i is an orbit of N, the ll:i's satisfy u1n = u1 for each n in N. Therefore~,·•·,~ span a subspace of ~(N). -50m Suppose w = r l l a 1kHgik lies in <11(N). Then i=1 k=1 m r l l (aikHgik - O'ikHgikn} = O i=1 k=l for each n in N. Now N acts as a transitive permutation group on each orbit, hence for each i, and each k,j such that 1 :$. i :$. m, and 1 ~ k,j :$. r, there is an element nikj of N such that If we use this relation in the sum above, as well as the fact that nijk permutes the cosets of Ci' it follows that, for each i, m Thus W = I a 1u1 , so the u1 span CM(N). Therefore, a Zp-basis for 1=1 M/U is the collection of cosets: u + Hg11 = u + H, u + Hg12'•••,u + Hg1,r-1 u + Hg21 (35) .• • • •• u + Hgm1 Suppose x lies in the kernel of the representation of G on M/U. Then, for 1 :$. i :$. m, and 1 :$. j :$. r - 1, . we have -51m Hgijx - Hgij = l ak'\: k=l where the ak are suitably chosen elements of ZP. Since Hgijx is ~ coset, and each '1c is a sum of at least three distinct cosets, we must have each ak = o. Since x permutes the ~rbits of N, it follows from the fact that x fixes Hgil that x fixes each orbit, (Ji. Therefore r-1 Hgirx = (ui - l Hgik)x = Hgir' k=1 so x lies in the kernel of the representation of G on the cosets of H. Therefore x = 1, hence M/U is faithful. Lemma 12. Suppose element, H, in ·V n e, 1 satisfy Hypothesis I, ec 1, and there is an e -°\"such that 0 (H) = 1. Then p 'lJ ~ {G Et I F(G) is a p-group). Proof. Let G be an element of' 1 such that F(G) is a p-group. Let Ebe an e-subgroup of G, and assume Op,(E'f) > 1. Since F(G) is a p-group, 0 ,(G) = 1, hence 10 (E )IG is faithful. p p' ~ Let M be the Zp(G)-module which gives the permutation representation on the cosets of 0 ,(E ); let U = n CM(O ,(E..r)g). p gEG p M1 = M/U, is faithful. By Lemma 11, the Z (G)-module, · p . -.52- Set X in = GMi,then F(X) = F(G)Mi,so X/F(X) ::::: G/F(G). Since G lies 1, X/F(X) E 'V, hence X lies in '/. By (2.5c) and Theorem .5, E* = E~ (Op,(~) is an e-subgroup of X. '(Op'(E ))=~(Op' (E~ )/U. jugates of E* in X. But ~(Op,(~)) 2 U, hence !st T be the intersection of all con- Since E* n M1 = ~ (Op' (&r')), it follows that 1 T n Ml = n ~ (0 • (E~g) = 1. gEX 1'11. p · But if K ~ x, and K n Ml = 1, then K:::: cx<M1>· ~(Ml) = CG(Mi)Mi_, Mi is self-centraX . li.zing in X, consequently, K = 1. Therefore T = 1, so 1E*I is so the fact that G is faithf'ul on M says that 1 faithf'ul. .,, It is clear, now, that 1E* IX is also faithful, so if t is any prime which does not divide the order of X, then XE {~t) = 1-t =V, by Isnuna 6. Therefore G, as a factor group of X, also lies in ti. · We may now assume Op,(Ei') to use H. If R = 1, so ~is a p-group. It is time = I 1 e •• • e It is a decomposition of the regular Zp (H)-module into its principal indecomposable constituents, we . set K.e = I.t /,/r1 for~= 1, • • •, t. Since R is faithful, and F(H) is a p'-group, it follows from Theorem 1 and Ismma 1 that R* is .f'aithful. = K1 e ••.~Kt. Since H does not lie in "r, it .follows that for some i, B = H/ CH{K,e) does not lie. in S. Ist K = K,e, then B is an element of °'Jn e- ~which has K as a faithful irreducible Zp(B)-modu1e. -53Let Ube the regular Zp (G)-module, and set W =(Bx G)(K ®U), where the action of B x G on K ® U by conjugation is the canonical action given in Chapter II. To show G E"lJ, it is sufficient to show W E iJ, sine~ G is a factor group of W. By (14), F(B) is a p'-group, and by assumption, F(G) is a Therefore if N is the kernel of the representation of p-group. F(B x G) on K ® U, then N = N n F(B) X N n F(G). Since B and G act faithfully on K ® U, K ® UI F(BxG) is faith.f'u1. By Lemm.a 1, K ® u is a faith.f'uJ. Zp(B x G)-module. Therefore op,(W) = 1, so F(W) = F(G)(K ® U). lies in Since W/F(W) ::: B x G/F(G), W/F(W) E 1J, so W 1. An e-subgroup of Bx G is Bx E, so by (25c) and Theorem ·5, E* = (B x E)CK ® 0 (op 1 ((B x E~) is an e-subgroup of W. B E '\Jn e-'I, 1 < ~ S F(B), so ~ is a p'-group. E'I is a p-group. Since By assumption . Let V be the collection of elements of B'r which appear as components of elements of (Bx E)~. Since (Bx E)-r~ Bx E, · V ~ B, and_it is clear that (B X Ehr S V X E. Therefore B/V ~ (B x E)!...V x E) lies in "r, hence V = Br• I f v E V, then there is an element u in E such that v X u E (B x El.r• Since ~is a p'-group, and Ei' is a p-group, v is equal to a power of v x u. fore (J6) There- -91Now K ® Uj ~is a multiple of Kj ~ so it follows from Theorem 1 that CK® u(Rr) = 1. By (36), B x Eis an e-subgroup of W. Let t be a prime which does not divide IWI. . w of Lemma 6, Wis an element of (~t) =1t = 'U. 1BxEI w is faithful. implies that 1(BxE~I The fact that . is faithf.Ul, so by part (b) Therefore Glies in V in every case, so Lemma 12 has been proven. Because of the preceeding lemma, we give Definition 7. · Set il = (pj Vne -'T contains a group, H, with o (H) p = 1}. We call a prime, p, special i£ p E il'. Lennna 13. If e, 1 satisfy Hypothesis I, and ec 1, then there is at most one special pl"ime. Proof. Let G be an element of minimal order in 1- e• Then G is the semi-direct product, EM, where E is an e-subgroup of G, and M is the unique minimal normal subgroup of G. Since E acts faithfully and irreducib~ on M, M= F( G). 10, Eis not an element oft, and since GE E E Vn By Lemma 1, G/F(G) lies in'V, hence £ - 'r. Since 0.r (E) n 0 s (E) = 1 for two distinct primes r,s, E/Ot.(E) can lie in 'T for at most one prime t. hence il' s; {t}. I f s # t, then E/O (E) E '\J s ne -'!', -55Remark. If general, we cannot control the choice of G enough to be certain that there are no special primes. This is the basis for the example in Chapter VI, and the reason behind liypothesis II. 1 - e... Let G = EM be a fixed element of minimal order in If r is any prime such that E/Or (E) lies in. 'T, we assume that Jr' is not contained in 1". Theorem 10. Suppose (Such a prime does not necessarily exist.) Eand 1 satisfy Hypotheses I and II. I f ec 1, then 1 = 1 , the collection of all solvable groups. Proof. Our first step is to show that 1J contains the collection, ill' of all solvable Tl-groups. By Lemma 12, the fact that 11.. s es "I shows tha.t '\J contains the collection of all nilpotent 'fl-groups. Proceeding by induction, we assume that °U contains the collection, '1..~ , of all solvable Tl-groups of nilpotent length at most i. But 'rt~+!= [GE~ jG/F(G) Ef'l~ }, thus 1 contains all solvable 11-groups of nilpotent length at most i + 1. Let G E '¥\.i;l, and F(G) = P1 X • • • x P8 , where P1 is the Sylow pi-subgroup of F(G). Set Ni= ~Pk, and let R1 be the regular Z (G/Ni)-modul.e for i pi = 1,•••,s. . We allow G to act on R = R1 X ••• X Rs by conjugation according to the ru1e -56_Then we form the semi-direct product X =GR. By construction, Ni centralizes the pi-group, Ri' hence the group, F(G)R, is nilpotent. Since F(X)/R is a normal nilpotent subgroup of X/R, and X/R::::: G, F(X) ~ F(G)R. Therefore F(X) = F(G)R, so X/F(X) z G/F(G). Since G/F(G) E 'V, it follows that X lies in 1. For each i, set xi= X/Ni(~Ric), Ri Gi = G(~1ic)/N1 (~l\:)• = NiR/Ni(~Ric), and By modularity Thus xi is the semi-direct product of Ri by Gi, hence Because 'G1 acts faith.f'ully on Ri' it follows that R1 is a self-centralizing normal pi-subgroup of xi. Therefore op~(Xi) = 1, so F(Xi) is a pi-group. But pi lies in TJ, so by Ismma. 12, Xi is an element of C\J for each i. Since n{Ni( TI~)}= 1, X lies in 'lJ also. i G E ~. Therefore k:/:i By induction, JTJ S 1.J. By Ismma 13, i f 1i.M is the minimal element of '/--e mentioned in Hypothesis II, then there is at most one prime, r 0 , such that E/O lies in 1, thus Tl contains r 0 •· Therefore, ro (E) -57- Suppose edoes not contain Jr~, and let G1 = E1~ be an element of minimal order in Jro• - e. By Lemma. 10, Ei is an element of 'V n {. - f, and since E1 E J.ro• , 0ro ( E1 ) = 1. . Therefore T] contains a1J. primes. Suppose l contains ) ro, • By assumption 'T does not contain J. • , ro so we can choose Hin Jro, .:.1. Since ,Sr• S 1J, H is an element of 0 1J n ~ - "f with. 0 ro ( H) = 1. Therefore Tl contains all primes in every case, so we have which completes the proof of Theorem 10. Corollary 2. at most i. Proof. Let?'l. be the collection of groups of nilpotent length . 1 If 11.1 << 1, then 1= f1 i, or 1 = j. I f we set ?1 = (1}, then for i 0 For each prime, p, ~ 1, 1l1 =(GE~ IG/F(G) Ef11 _ 1 ). Ap' is not contained in ?f1 _ 1 , hence the hypothesis of Theorem 10 is satisfied. r The result follows from Theorem 10. -58Chapter VI An Eicample · Let r be a prime. Throughout this chapter, we shall set ~= {G EJIG/F(G) E ,&r,}• For each p, we set ~p = ,&r,; (~) forms a local definition for ~ by Lemma 9. In this chapter, we shall characterize the formations which strongly contain~. The for- mation ~ provides an example which shows that Hypothesis II is not redundant. Lemma 14. Let G be a group with Sylow r-subgroup R. Then NG(R) is an~-subgroup of G. Proof. Because R is a Sylow r-subgroup of G, NG(R)/R is an r'-group. Therefore NG(R) lies in ~. Suppose NG(R) ;S U ~ G. We wish to show NG(R) covers U/Ua,• Clearly UCP-_ = 0r' ' r (U). Since R ~ 0r' (U), it follows that 0r' (U) .a cts transitively on the Sylow r-subgroups of U, hence U ~ 0r' (U)NG(R). Since R covers every r-section of U, it follows that NG(R) covers 0r' (U)/Or' ' r (U). Therefore NG(R) covers U/ 0r'r ' (U). By definition, NG(R) is an ~-subgroup of G. Suppose 1 >>&, and 1;:,~. I f £1.q } is the minimal local def'in- ition of 1, it follows from Theorem 9 that 1 = 1 for each q,s.· . q s We set dlJ = 'f.q • I f H lies in "IJ n ~ , then H has a normal Sylow -59- r-subgroup, so H/Or (H) lies in~ r ,. violated for the prime r. Therefore, Hypothesis II is It follows from lemma. 13 that r is the unique special prime associated with 1 and~. The next theorem gives a cl.ass of formatiohs which strongly containd<. Theorem 11. I.st"\ be a non-empty formation. I.st "U= (GEA \G/Or ,(G) E'I}, then '1J is a formation. If ·1= (GEA IG/F(G) E'U}, then "J>>@... Proof. Suppose G E°U, and N ~ G. Then Or 1 (G)N/N .:S: Or,(G/N). Since G/Or' (G) E j', we also have (G/N)/Or' (G/N) E °I. Therefore, G/N lies in ~. Suppose N1 ,N are two normal subgroups of G such that G/Ni E~ 2 for i = 1,2. For each i, let Mi/N1 = Or 1(G/N1 ), then G/Mi E "r for each i, hence G/Mi n ~ E "f. For each i, (M1 n ~)N1 /Ni is an r'-group, hence M1 n ~/Ni n Mi n ~ is an r'-group for each i. Therefore M1 n ~/N1 n N2 is an r'-group, so it follows that the factor group of G/N1 n N2 by Or' (G/N1 n N2 ) lies in 'f. Th~refore G/N1 n N .E 1J, thus iJ is a formation. 2 To show CX<<'1, it is sufficient to show that ~PS 'tJ for each prime p. Suppose G E fr' then G has a faithful. irreducible Zr (G)-modu1e. By (14), G/F(G) E\). Since Or(G) this means that 0r (G) = 1. Since G E 'f, = 1, F(G) is an r'-group, so G/Or (G) E '1. 1 -6oTherefore G lies in 'lJ. Suppose G E ~p for p F r. R is a Sylow r-subgroup of G. .An ~-subgroup of G is NG(R) where Since p # r, 0 ,((NG(R))_g r' p ) = R. Therefore G has a faithful irreducible Zp (G)-module, J, such that 1 < CJ(R) .::;:: J. identity. By Theorem 1, either CJ(Or(G)) = J, or it is the The latter possibility cannot occur because 1 < CJ(R).::;:: CJ(Or(G)). Therefore the fact that J is · faithful says that Or(G) = 1, so F(G) is an r'-group. G E1, so the same argument as in the preceeding paragraph shows that G/Or 1 (G) E°f'. G E tJ. By Theorem 6, !'.K<< Therefore 1. Since our choice of 'I is arbitrary, it follows that we can . choose an infinite number of distinct formations which strongly contain fR. Our next theorem is the converse to Theorem 11. Theorem 12. Suppose ition for 1. Then there is a non-empty formation °I such that 1 >> ~ , and £1q J is the minimal local defin- ; '1-q = (G El IG/Or' (G) E 'fJ. Proof. Suppose '1=>~. By Theorem 9, there is a formation 'lf such =1J for each q. Our first step is to show that 6lJ is the that 1 q smallest formation generated by {H E"f lor(H) = 1}. Let 'tJ' be the smaJ.lest formation generated by this set. Sup:Pose HE 7, and Or(H) = 1. Let K ~ I 1 ~ ••• e Is be the decomposition of the regular Zr(H)-module, K, into principal -61indecomposable submodules. By Theorem 1, Lemma 1, and the fact that F(H) is an r'-group, it follows that H acts faithfully on J 1 @ ••• ~ Js' where Jk ~ = H/CH(Jk). = Ik~~ for each k. For each k, set Then Jk is a faithful irreducible Zr(Hk)-modu1e. If~ is a Sylow r-subgroup of~' then NHic(~) is an!R-subgroup of Hie' and (NHic(~)~r' =(NI\:(~))~=~· Or,((NHic(~))Qr) ~ << = 1. Therefore B-3 definition, Hie E tr for each k, so t implies Hie E 1r = lJ • . Therefore H lies in V. Since the generators of 1)' all lie in 'U, ~ is contained in(]. We know that ..· '\J is the smallest formation generated by t , from the proof of r ,..,, ,._ Theorem 9. Thus if we show tr S 'LJ , we have shown 'ti S LJ. If G E tr, then G has a faithful irreducible Z (G)-modu1e, and G E r 7. By (14), = 1 so by definition, G E tf. Therefore lJ = '1. Let 'I' be the smallest formation generated by the set (H/Or,(H)IH E1J}. Set 111 = (G El IG/Or,(G) Eer}. We want to show Or(G) .r 1) =1)1 • By construction, 1) S 1) • 1 Since the generators of l1f are elements of d)), we must have "f £ l). Therefore, i f G E iJ 1 , then G/Or' (G) lies in 'U • .To show G lies in'U, we use induction on the nilpotent length of Or,(G). 0 , (G) . is nilpotent, then G/O , (G) E r G is an ele~ent of r 1. 'U forces G/F(G) EV. Therefore By our first paragraph, G/Or(G) lies in · G also lies in~ since Or(G) n or,(G) = 1. I£ 'U, so -62- We note that 0 ,(G/F(O ,(G))) = 0 ,(G)/F(O ,(G)), hence by r r r r induction, if GE 4\) 1 , then G/F(Or 1 (G)) E~. 1. Therefore G lies in By our first paragraph G/O (G) lies in ILJ, so G/O (G) n F(O ,(G)) • G r r r also lies in \I. Therefore the case when Ole This completes the proof in 1. If~--= 1, we let l identity. V= tJ1 • be the fonnation consisting only of the We must then show that {~} is the minimal local defin- ition for~. Let {(jl q } be the minimAl local definition for IR.. an arbitrary prime, G E ,8 , r divide rplGI. Suppose p is =(R_P, and t is a prime which does not Let K be the regular Zt(G)-module. Let K be the regular ZP(G1 )-module. 1 Set G1 =GK. Set G = G1 K • 1 2 Since G acts faithfully on K, and G1 acts faithfully on K1' op,,p(G2 ) = K1 • Depending on the choice of p, G is either an r'-group, or has K 2 1 as a normal Sylow r-subgroup. GiO , (G ) = G /K E ( • 2 1 p p ,p 2. follows that G E Olp• Therefore G E ~' hence 2 Since G is a factor group of G1 , .it . Therefore ,gr' ~ [p. This completes the proof. -63- Appendix In this appendix, we shall give proofs to (18'), (18), and (19). (18') If N ~ G, and n 1 ,•••,nn are sets of primes, then Proof. Use induction on N. If n = 1, and TT = n 1 , then On(G) is the unique maximal normal TT-subgroup of G, hence 0TT (G) is .characteristic Therefore, 0TT (N), as a characteristic subgroup of the normal in G. subgroup N, is normal in G. Thus OTT (N) -< 0TT (G) n N. But N n 0TT (G) is a normal n-subgroup of N, hence is contained in 0TT (N). We may assume that T = O (N) = O (G) n N. TT1,•••,TTn-1 TT1,•••,TTn-1 Then 0 (N)O (G)/O (G) is a norma1 n1,•••,nn n1,•••,TTn-1 TT1,•••,TTn-1 · TT -subgroup of G= G/O ••• (G), and must be contained in n TT1' 'TTn-1 O (G). TTn Therefore O (N) < O ••• (G) n N. n1,•••,TTn - TT1' ,nn But 0 ••• (G) n N/T is a normal TTn-subgroup of N/T, hence TT1' ,nn s OTT ••• TT (G) n N OTT ••• TT (N). This proves (18'). 1' ' n 1' ' n (18) . Let G be a group. Then 0 p , ,p(G) = n CG(K), where 'the intersection is taken over all p-chief factors, K, of G. Proof. By looking at a chief series of G which passes through Op , ,p (G), it is clear that 0p , ,p (G) centralizes every p-chief factor of G. Let P = 0p , ,p(G)/Op ,(G), and E P/D(P). Then G is represented = -64on E, and the kernel of this representation is 0 , (G) (cf. [9],p. 7). p ,p Ii' g is a:ny p'-element of n CG(K), then g centralizes E, since it centralizes every p-chief factor of G. Therefore g lies in Op,(G). This shows that n CG(K) has a normal p-complement, so by (18'), n CG(K) :s op,,p(G). (19) If G is a group, then F(G) = n 0 , p p ,p (G), where the intersection is taken over al1 primes p. Proof. Since F(G) is nilpotent, it has a normal p-complement for By (18'), F(G) ~ n 0 , every p. (G). Therefore it is sufficient (G) is nilpotent. But look at a:ny chief series . to show that n 0 , ' p p ,p p p ,p of G passing through n O , (G). . p p ,p By (18), n O , p p ,p (G) centralizes every factor in this series, hence the part of this. chief series covered by n O , ( G) forms a central series for n 0 , . ( G), . p p ,p p p ,p so n 0 , (G) is nilpotent. p p ,p This proves (19). -6.5References 1. Artin, E., Nesbitt, c. J., and R. M. Thrall, RINGS WITH MINIMUM CONDITION. University of Michigan Press, Ann Arbor, 1944. 2. Carter, R. W., "Nilpotent self-normalizing subgroups of soluble groups." Math. Zeit., 7.5(1961), pp. 136-139. 3. CUrtis, W., and I. Reiner, REPRESENTATION THEORY OF FINITE GROUPS AND ASSOCIATIVE AIJJEBRAS. Interscience, New York, 1962. 4. Fong, P., and W. Gaschfltz, "A note on the modular representations of solvable groups." J. ffir Math., 208(1961), pp. 73-78. .5. Gasch'f:ltz, W., "Zur Theorie der endlichen aufl8sbaren Gruppen." Math. Zeit., 80 (1963), pp. 300-30.5. 6. Gaschfitz, W., and U. Lubeseder, "Kennzeichnung gesllttigter Formationen." Math. Zeit., 82(1963), pp. 198-199. 7. Glauberman, G. "Fixed points in groups with operator groups." Math. Zeit., 84(1964), pp. 120-12.5. s. Green, J. A., "On the indecomposable representations of a finite group." Math. Zeit., 70(19.59), pp. 430-44.5. 9. Hall, P., and G. Higman, "On the :irlength of :irsoluble groups and reduction theorems for Burnside's problem." Proc. London Math. Soc. (3)6(19.56), pp. 1-42. 10. Lubeseder, u., Unpublished Thesis. ·Germany. University of Kiel, Kiel