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Embedding theorems in finite soluble groups Hughes, Peter Walter 1971

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Embedding Theorems i n F i n i t e Soluble Groups by Peter Walter Hughes B. S c , U n i v e r s i t y of Auckland, New Zealand, 1968 •M.Sc, U n i v e r s i t y of Auckland, New Zealand, 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF S€*EWeE AfCTS In the Department of MATHEMATICS . & We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1971 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y of B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f The U n i v e r s i t y o f B r i t i s h C o l umbia Vancouver 8, Canada Date Uf[. If 7/. By a group we will mean a finite soluble group. It is an interesting fact, (Pardoe [1]), that the subgroup closure of the class of groups P^ , those with a unique complemented chief series, is a l l groups. Let X be the class of groups with a complemented chief series. We investigate the action of closure operations T such that TX = X upon P^ . The purpose of this is to find a collection of such closure operations whose join applied to is X . In the course of this investigation we introduce a new closure operation M defined by ; M/ = { G | G = <X1}--.,Xn>, X ± e. V, X ± sn G, ( |G : X j J . - . j G : X n| ) = 1 } ACKNOWLEDGEMENTS The Author i s indepted to h i s supervisor, Dr. Trevor Hawkes f or the suggestion of the topic of t h i s t h e s i s , and f o r his valuable assistance and encouragement throughout i t s preparation. He would also l i k e to thank Dr. B. Chang f o r reading the draf t copy. He i s g r a t e f u l f o r the f i n a n c i a l assistance given by the Canadian Commonwealth Scholarship and Fellowship Administration. F i n a l l y i t i s a pleasure to acknowledge the patience, the care and the p r o f i c i e n c y of Mrs. Y.S. Chia Choo i n typing t h i s t h e s i s . Introduction 1 Chapter 1 §1.1 : D e f i n i t i o n s and Notation 3 §1.2 : Some Known Results 10 Chapter 2 §2.1 : Pr e l i m i n a r i e s 30 §2.2 : Two Closure Operations 33 §2.3 : A Further Investigation of X ' 37 Conclusion 46 Bibliography 47 Introduction When given a group G with a property (*) i t is often useful to know whether subgroups H of G, or quotient groups G/K of G, also possess this property. We can pose this question in a slightly different manner. Consider the class of groups G with (*)> what properties characterise the subgroups, or quotient groups, of these groups ? The answer to this question gives us a deeper understanding into the structure of the groups with (*). Of course we may also ask questions about direct products, or subnormal subgroups, of groups with (*). The object of this dissertation is to pose a question of this type i n a formal manner and give some answers for certain properties (*). More e x p l i c i t l y , given a class of groups X and a closure operation T, is there a simple description for the class TX ? One way to tackle this problem is by looking at T-closed classes of groups V that contain X and deciding whether or not, TX = Y. An obvious extension of this method i s to look at the join of several such closure operations acting on X . By a group we w i l l always mean a f i n i t e soluble group and thus our universal class of groups w i l l be the f i n i t e soluble groups. We w i l l denote this class by S. In Chapter 1 we w i l l discuss some known results of the type discussed. One measure of the complexity of a class of groups X relative to 5 is to look at i t s subgroup closure. If sX = S then we can say X i s " l a r g e " i n as f a r as a l l groups can be embedded i n an X-group. Chapter 1 contains s e v e r a l r e s u l t s of t h i s s p e c i a l type f o r widely d i s s i m i l a r classes X. In Chapter 2 we discuss two s i m i l a r classes of groups, those with a unique complemented chief s e r i e s and those with a complemented chief s e r i e s . We i n v e s t i g a t e whether the former can be extended to the l a t t e r by a j u d i c i o u s choice of closure operations that leave the l a t t e r f i x e d . As yet t h i s question has not been answered, but some progress has been made towards s e l e c t i n g a s u i t a b l e set of closure operations that may work. §1.1. D e f i n i t i o n s and Notation. One of the fundamental tools used to generate new groups from known groups i s the wreath product. There are several d e f i n i t i o n s f or the wreath product. The version we w i l l need the most i s the regular wreath product and t h i s i s defined as follows. D e f i n i t i o n : The regular wreath product, G wr. H, of a group G by a r group H, i s defined by ; G wr. H = { (f,h) | h e H, f : G — > H} r where m u l t i p l i c a t i o n i s defined by ; ( f 1 , h 1 ) ( f 2 , h 2 ) = (g,h 1h 2) , and g(h) = f 1 ( h ) f 2 ( h h 1 ) f or a l l h i n H. As we w i l l u s u a l l y be considering the regular wreath product we w i l l drop the s u f f i x 1 r * . We w i l l need several r e s u l t s about wreath products. One used i n (1.2.9) i s concerned with extending an automorphism of G, (or H), to an automorphism of G wr. H. We also need two embedding properties that are used extensively i n what follows. (1.1.1) Lemma : Huppert [1]. (a). I f a i s an automorphism of G, so i s cT an automorphism of G wr.'H where ; ( f , h ) " = (g,h), and g(k) = f ( k ) a f o r a l l k i n H. (b). I f u i s an automorphism of H, so i s u an automorphism of G wr. H where ; ( f , h ) y = (g,h y), and g(k) = g(ku _ 1) for a l l k i n H. (1.1.2) Lemma : Huppert [2], (a). If G-^  <_ G and % <_ H then G^ wr. H^ can be embedded i n G wr. H. (b). Let N«4 G, then G can be embedded i n N wr. G/N. Theorem (1.2.11) provides not only a r e s u l t of i n t e r e s t but also a method to construct groups that have a unique complemented chief s e r i e s . This construction i s l a t e r used i n Chapter 2. In the proof we use two general r e s u l t s which we w i l l quote here for completeness. The f i r s t r e s u l t allows us to c a l c u l a t e the order of c e r t a i n c h i e f f a c t o r s , while the second r e s u l t i s used to provide a co n t r a d i c t i o n i n the proof of (1.2.11). (1.1.3) Theorem : Huppert [3]. Let V be a f a i t h f u l i r r e d u c i b l e module of dimension n over the f i e l d K = GF(p f), f or an Abelian group A. Then A i s c y c l i c and there e x i s t s a group homomorphism ; 3 : A > G F ( p n f ) * , and a K-isomorphism, a : V > G F ( p n ^ ) + such that, a(va) = 8(a)a(v) f o r a l l a e A and v e V. The integer n i s the uniquely determined smallest integer such that | A | p n f - 1 . (1.1.4) Theorem: Huppert [4]. Let G<_GL(n,p f), K = GF(p ), and A an Abelian normal subgroup of G. Let V = V(n,p n^), considered as a K[A] module, be the d i r e c t sum of s = n/k i r r e d u c i b l e isomorphic K[A] modules. Then G, considered as a permutation group on V, acts i n the same way as a group of GF(p^) semilinear maps g, such that ; ( u 1 + u 2)fe = u-jg + u 2g, u i e V(s,p (uc)g = ugcS, u e V ( s , p k f ) , c e GF(p ) , where c — > c& i s a f i e l d isomorphism. Also C^(A) I s t n e s e t °f GF(p k f) - l i n e a r maps of V ( s , p k f ) . (1.1.5) C o r o l l a r y : G/C r(A) i s isomorphic to a subgroup of C k ' Proof : Consider the map a : g -> a where a c — > c g That i s , g a G -> Aut [GF(p k f) : GF(p f)] = Galois group [GF(p k f) : GF(p f)] Also a i s a homomorphism with kernel given by ker a = {g e G | a =1} = {g e G | c 8 = c f o r a l l c i n GF(p k f)} = CG(A) . Hence G/Cg(A) = a(G), a subgroup of . Another r e s u l t used i n the proof of (1.2.11) i s a theorem of D i r i c h l e t ' s concerning the d i s t r i b u t i o n of prime numbers i n a given sequence of numbers. (1.1.6) Theorem : D i r i c h l e t [1]• I f (m,n) = 1 then the s e r i e s {m + kn | k = 1, 2, •••} contains an i n f i n i t e number of prime numbers. We w i l l now introduce the concept of group classes and closure operations. These concepts are extremely u s e f u l and w i l l be used throughout t h i s exposition. We say a set of groups X defined by some'group th e o r e t i c property possessed by a l l i t s members i s a c l a s s of groups i f ; (a) . I f G e X, then a l l groups isomorphic with G are also i n X, (b) . The u n i t group 1 i s i n X. We w i l l denote by (1) the t r i v i a l class of groups and by (G) the class of groups isomorphic to G, (with 1). A closure operation c i s a map from classes of groups to classes of groups s a t i s f y i n g ; (C.l).. X <_ CX, expanding, (C.2). I f X <_Y then CX <_ CV, monotonic, (C.3). CX = C(CX), idempotent. We say X i s C-closed i f CX = X. I f B and C are closure operations then C <_ B i f and only i f for a l l classes of groups X, CX <_ BX. The j o i n {A,B} of two closure operation A and B i s defined by, {A,B}X = n {y | x < y, AY = B/ = y} . By AB(X) we mean A(BX) and we note that AB i s a closure operation i f and only i f BA <_ AB. An alternative method to describe a closure operation C i s to specify the C-closed classes. Let C be a closure operation and C = {X ( CX = X} . Then the family C s a t i s f i e s ; (a) S e C (b) C i s closed under taking arbitary intersections. Conversely i f C i s a family of classes of groups s a t i s f y i n g these two conditions then we may define a closure operation T and i t s action on an arbitary class of groups X by ; TX = n {y | x <_ y e C} . Thus there i s a one to one correspondance between closure operations and families of group classes that s a t i s f y (a) and (b). The following can be shown to be closure operations. SX = { H | H <_ G E X} . QX = {G/K | K«d G e X} . SnX = { H | H sn G e X} . NQX = {G | G = <NX, Nr>, N ± £ X, N ± sn X} . DQX = {G | G = H X x ••. x H R , H ± e X) . r RDX = {G | 3 G, G/N± e X, H N ± = 1, i = 1, • • •, r>. i=l Also the common classes of nilpotent and Abelian groups w i l l be denoted by W and A, respectively. Notation : If G is a group and H is a subgroup of G we define ; H <_ G, H is a subgroup of G. G, H is a normal subgroup of G. G, H is a minimal normal subgroup of G. H <3 oG, H is a maximal normal subgroup of G. H sn G, H is a subnormal subgroup of G. Core H , largest normal subgroup of G contained in H . N Q ( H ) , the normaliser of H in G. C G ( H ) , the centraliser of H in G. Z(G), the centre of G. F ( G ) . the F i t t i n g subgroup of G . <)>(G) , the F r a t t i n i subgroup of G . A u t ( G ) , the group of automorphisms of G . <A>, the subgroup of G generated by a subset A of G . = <H§ | g e G > , the subgroup generated by the conjugates of H i n G . [g»h] = g ^ n ^gh> the commutator of g and h i n G. [H,K] = < [h,k] | h e H, k e K >. G * = [ G , G ] , the commutator subgroup of G . | G | , the order of G . J G : H|, the index of H i n G . I T (G ) , the set of prime d i v i s o r s of G . G ^ , a H a l l Tr-subgroup of G , where TT i s a set of primes. i . e . a larges t subgroup of G whose order i s a Tr-number. Cp , the c y c l i c group of order p. S(n), the symmetric group of degree n. If a and b are integers we define, a|b, a divides b. a|b, a does not divide b. (a,b), the greatest common d i v i s o r of a and b. The following explains the notation used i n (1.1.3), (1.1.4). Zq, the integers modulo q, the f i e l d of q elements. GF(p n), the Galois f i e l d of order p n . GF(p ) , the a d d i t i v e group of the f i e l d . GF(p n)*, the m u l t i p l i c a t i v e group of the f i e l d . GL(n,pf), the group of i n v e r t i b l e l i n e a r transformations of a vector space of dimension n over GF(p^). §1.2. Some Known Results. In t h i s s e c t i o n we w i l l b ring together some known r e s u l t s about closure operations a c t i n g on given group cla s s e s . I t i s w e l l known that the converse of Lagrange's theorem i s f a l s e . There are groups, f o r example, the a l t e r n a t i n g group of degree 4 , which do not possess subgroups of a l l possible' orders d i v i d i n g |G|. I f M denotes the cl a s s of groups which have subgroups of a l l p o s s i b l e orders i t i s c l e a r from P. H a l l ' s s u f f i c i e n t condition f o r s o l u b i l i t y that M <_S. The following theorem shows that, i n the sense that any soluble group can be embedded i n a group i n M, that the cl a s s M i s large i n r e l a t i o n to S. (1.2.1) Theorem : Maclain [1]. S = sM Proof : Let G be a soluble group with order II p. . Let U be an i i = l 1 r a . - l Abelian group of order II p. then G x U i s i n M. For, l e t i = l 1 d l d2 d r . . d = p^ p 2 P r be a d i v i s o r of [G x U|. We can reorder the d / s such that <^  d^ , i = 1, 2, • • •, s ; d^ < , i = s + 1, • • •, r. Let H be a -Hall iT-subgroup of -G where TT = {p^, p 2 , '*", Pg}> a n ^ B a d l ~ a l d s " a s d s + l d r subgroup of U of order p.. ••• p p . ••• p . Then |H X D| =d J_ S S T I and the proof i s complete. A subgroup H of G i s c a l l e d complemented i f there is a subgroup K of G such that, HK = G and H f\ K = 1. The class of groups C i n which every subgroup has a complement, i s c a l l e d the c l a s s of complemented groups . We have C <_S, by P. H a l l ' s c h a r a c t e r i s a t i o n of soluble groups, namely that G is soluble i f and only i f i t has Sylow p-complements for a l l prxmes p | G | . The class C i s both S- and D 0~closed. Also i f G e C then the Sylow p-subgroups of G are elementary Abelian and the chief f actors of G are c y c l i c . A subclass of C i s the class of groups Q, the groups of squarefree order. That these groups are soluble follows from Burnside's Theorem, namely that i f a Sylow p-subgroup i s contained i n the centre of i t s normaliser then i t has a normal Sylow p-complement. Further, i f H <_ G e Q then | G | = | H | m where ( |H|, m) = 1. So, by H a l l [1], H has a complement i n G . We now turn to the connection between Q and C. (1.2.2) Theorem : P. H a l l [3]. SD Q(Q) = C, Proof : By closure axiom (C.2) and the above remarks SD Q(Q) <_ SDQ(C) = C. Thus we need only show the opposite i n c l u s i o n . Let 1 ^ g e G and l e t G be a normal subgroup of G maximal with respect to not containing g. Then G i s isomorphic to a subgroup of the d i r e c t product of the groups Hg = G/Gg. We know by the above remarks that a group G i n C has the following p r o p e r t i e s . (1) Every Sylow subgroup of G i s elementary Abelian. (2) Every chief f a c t o r of G i s c y c l i c . The groups H g have the properties (1) and (2) and also the property. (3) of containing a unique minimal normal subgroup. Thus i t i s s u f f i c i e n t to show that a group G s a t i s f y i n g (1), (2), and (3) i s i n Q. By (2) the unique minimal normal subgroup N of G i s c y c l i c and has order p. Let C = C G(N). We show C = N. Suppose not. Then there e x i s t s an Abelian normal subgroup K of G, such that N <_ K <_ C, and K/N i s a chief f a c t o r of G. By (2), K/N i s c y c l i c of order a prime q. (a). Suppose p ^ q. Then, K being Abelian, the Sylow q-subgroup of K i s c h a r a c t e r i s t i c i n K, and G has a minimal normal subgroup d i f f e r e n t from N, con t r a d i c t i n g (3). (b). Suppose p = q. Then, as |K | = p^, K < G, and as each Sylow p-subgroup of G i s Abelian by ( i ) , K i s c e n t r a l i s e d by each Sylow p-subgroup of G. So c e r t a i n l y N i s c e n t r a l i s e d by each Sylow p-subgroup of G and thus F = G/C G(N) i s a p' group isomorphic to,a subgroup of Aut(N). So we can form w = K ] F , the semidirect product of K by F , in. the n a t u r a l way. As N^J G and ( | K | , | F | ) = 1 and K i s Abelian we can use Maschke's theorem to give K = N x M, where M<J G. This again contradicts (3). Thus N = C and G/N i s isomorphic to a subgroup of Aut(C ), which i s c y c l i c of order p - 1. i . e . G i s metacyclic of order pk, where k 1. By (1), G contains no c y c l i c subgroups of order the square of a prime ; so k i s square free and G e Q., completing the proof. Nilpotent groups are characterised by being the d i r e c t product of t h e i r Sylow subgroups. Thus i t i s always possible to construct a normal s e r i e s i n which the f a c t o r groups have order the lar g e s t power of each prime d i v i s o r d i v i d i n g the group order. Thus n i l p o t e n t groups form a subclass of the c l a s s of Sylow Tower groups T, which i s defined by, T = { G | G has a normal s e r i e s 1 = GQA G-, <d. •••<<! G n = G where the normal factors G±/G±„j_ a r e isomorphic to Syl subgroups of G} . ow A normal s e r i e s for a group G may be defined by 1 4 0 p(G)<3 0 p I p ( G ) 0 , (G)'<3 •••<G (1) where if IT is a set of primes G^ CG) is the product of a l l the normal subgroups of G whose orders are divisible only by members of TT. We define 0^  w v (G), where TT . , i = 1, r are sets of primes, r r-1' 1 inductively by • r-1 A = 0_ G/°Trr_1...Tr1(G) The set of primes I T ' is just the complementary set of primes to TT . Clearly for a nilpotent group the series stops at 0ptp(G) = G. That is the number of factor groups in (1) divisible by p for each prime p is one. However this is also true for Sylow'Tower groups as the series reaches G at most at °p'pp' ( G)• Groups with this property of having only one factor group in the series (1) divisible by p for a l l primes p are called groups of p length 1, and form the class Lm(l). The class L ^ ' C l ) can also be locally defined by the formation function f : p —> S^,. The class S^T of a l l groups not divisible by p is {Q, S, RQ, NQ} -closed and these closure properties are inherited by the locally defined class 1.^(1). For a description of formations and formation functions see Huppert [5]. (1.2.3) Theorem : Alperin [1]. SN T = Lm(l) Proof : It is clear that T <_ 1^(1) and so by closure operation axiom C.2, SNQT <_ SN0Loo(l) = L o o(l). To show the opposite inclusion we prove the stronger result namely : I f G e L r a ( l ) then, ( 1 ) G = H ^ L ^ - ' - I ^ = L where L ±<4 L and L± e T, and ( 2 ) T r ( L I ) <_ T T(G ) . We prove t h i s a s s e r t i o n by induction on | G | . (a) . I f N-^  and N 2 are two d i s t i n c t minimal normal subgroups of G then, as Is w e l l known, G/N-^ * G/N 2 contains a subgroup isomorphic with G . So by the inductive hypothesis G e S D 0 ( S N 0 ( T ) ) = SN Q T , and T T C G / N - L ) U T T ( G / N 2 ) = T T(G ) . (b) . Suppose G has a unique minimal normal subgroup N. If {p} = TT(N), then 0 P ? ( G ) = 1 , and P the Sylow p-subgroup of G i s normal i n G . So by the induction hypothesis ; ( 1 ) . G / P = H <_ L-jL 2* • - L N e N Q T and ( 2 ) . -n(.L±) <_ T T ( 6 / P ) . So., P wr. G / P <_ P wr. L = B L = B L ^ B L 2 • B L N E N 0 T , where B i s the base group of the wreath product and by ( 2 ) p/ | L | . Thus T T C B L ^ X £ TT(G) and by ( 1 . 1 . 2 ) , G e SN D T completing the proof. Nilpotent groups have the property of being both S n - and N Q-closed. In f a c t , classes of groups which are closed under these two operations are quite numerous and have been studied to determine properties common to them a l l . Thus we say a class X i s a F i t t i n g c l a s s i f X =' {S n,N 0 , }X . We w i l l c a l l a F i t t i n g class F t r i v i a l i f F = ( 1 ) or S . For any class of groups X and any group G we define a subgroup V of G, called the X-injector ; i f for a l l N sn G, N (\ V i s X-maximal. i.e. N H V e X and is not contained properly in any X-subgroup of N. Of course such a V may not exist, but what x^ e can say i s , i f F is a Fitti n g class, then every group has an X-injector. It was proved by Gaschutz, Fischer, and Hartley [1], that when X is a Fitting class, every group has a unique conjugacy class of X-injectors. It now makes sense to c a l l a F i t t i n g class F normal i f for any group G the F injector Gp of G is normal i n G. normal F i t t i n g classes and give a theorem of .the type we are discussing. What follows i s a summary of some -of the-results ©f this paper. F i r s t we w i l l give an example of a non t r i v i a l normal Fitting class. In a recent paper, Gaschutz and Blessenohl [1], investigate (1.2.4) Example : G-B ; Satz 33. Consider a group G with chief series C and let •••, Mr be the p chief factors in C. Let A be a cyclic group of order p - 1. We define d-i(g) = determinant of the linear transformation induced on by conjugation with g. r Let d r(g) n d ±(g) i=l or d G(g) = 1 i f G has no p-chief factors. Define for a l l g in G } . Then the classes F for each prime p are non t r i v i a l normal Fitting classes. If a non trivial Fitting class F contains a group G and p a prime, then there exists some n such that mnG wr. C p is in F, a l l integers in > 0. For any class of groups X we define the .'-.'.'/'racteristic of X by •char X = { p I C e X } . P F is a Fitting class this definition is equivalent to defining char F = {p | there is a G in F such that p |G|} il-'Us a normal Fitting class F ± 1, contains the class S p of p-groups !:V.i- a l l primes p. But the nilpotent groups W are simply direct products p-groups, and hence N <_ F, for a l l normal Fitting classes F ^ 1 . A non trivial normal Fitting class is a large class of groups, in the ;^!)se that its subgroup closure is a l l soluble groups. \.|.; 2.5) Theorem : Let 1 ? F be a normal Fitting class then sF = S . -*L»;gof : We may assume that F < S. Let G e S and proceed by induction |G| . Let Gx^3 oG and |G : G j = p. If G-j^  = 1, then G e W <_ F . ^ G-. ^ 1 then there is a m such that mGiwr.C e F . But then there ^c-iHist monomorphisms y and y 2 such that v 1 y 2 G > G-^ wr. G/G-j^  = Gj wr. C p > m Gj^ wr. C p In general normal F i t t i n g closses are not Q-closed. This i s shown i n the f o l l o w i n g example. (1.2.6) Example : Let F^ be the non t r i v i a l normal F i t t i n g c l a s s described i n (1.2.4) with p = 3. Let G be the semidirect product of an elementary Abelian group of order 4 with an i n v e r t i n g i n v o l u t i o n , i . e . G = AX where A = x C^ and |x| =2. We can consider elements of G to be t r i p l e s (a,b,y) where (a,b) e A, y e X; and m u l t i p l i c a t i o n i s defined by ((a,b),'y)-((u,v).,z) = ((a,b) ( u , v ) y , z ) . Then G has a c h i e f s e r i e s 1 4 1 < ( ( a , l ) , l ) > = B 4 B « < ( ( l , b ) , l ) > = A<d G- = AX. Let = B, M 2 = A/B, then the M.^  are the 3-chief f a c t o r s . Now d i(g) = 1 i f g i s of the form (( u , v ) , l ) and d^(g) = 2 i f g i s of the form ((u,v),x). In a l l cases d G(g) = d-i(g) X d 2(g) = 1 and G e F^. But Sg = G/B and the chief s e r i e s f o r i s l«d A ^ ^ S^ , with 3-chief f a c t o r A 3 = M^. In t h i s case dg(g) = d-^g), and d 1((12))=2. So S3 i ^2 • This example suggests the question, "what i s QF f o r F a non t r i v i a l F i t t i n g c l a s s ?" I t would be of i n t e r e s t to know whether, f o r such F, QF i s the u n i v e r s a l c l a s s . If F i s a F i t t i n g Class define the clas s Radical-Quotient (F) = = { H | H = G/Gp f o r some G} . Blessenohl and Gaschlitz prove that i f F i s a non t r i v i a l normal F i t t i n g c l a s s then Radical-quotient ( F ) <_ A . In other words the F - i n j e c t o r s of G always contain the derived group G' . Moreover, they show that t h i s s t r u c t u r a l r e s t r i c t i o n on Radical-quotient ( F ) characterises normal F i t t i n g c l a s s . Indeed, any s t r u c t u r a l r e s t r i c t i o n on Radical-quotient ( F ) seems to imply normality of the F i t t i n g class F i n the following precise sense : (1.2.7) Theorem : B-G. [1] Let X be a S-closed class of groups d i f f e r e n t from S . Suppose F i s a F i t t i n g c l a s s and F <_X.. Then F i s a non t r i v i a l normal F i t t i n g c l a s s . (1.2.8) Theorem : F.P. Lockett. [unpublished]. Let X be a Q-closed class of groups d i f f e r e n t from 5. Suppose F i s a F i t t i n g c l a s s and F <_X. Then F i s a non t r i v i a l normal F i t t i n g c l a s s . In other words, i f F i s a non normal F i t t i n g c l a s s , then C(Radical-quotient ( F ) ) = S, f o r C = S or C = Q. In any soluble group the Sylow subgroups always have Sylow complements and these are conjugate, and any subgroup of order prime to a Sylow p-subgroup i s contained i n a Sylow p-complement. P.Hall [4] has shown that these properties can be c a r r i e d over to Sylow systems. Let S]_, ••*,S be a complete set of Sylow-complements i n a group G of order p • ''T?r ' i ' e * One representative from each of the r conjugacy classes of complements of Sylow subgroups. A Sylow system consists of a l l 2 r i n t e r s e c t i o n s , i n c l u d i n g the empty i n t e r s e c t i o n G, formed from these r subgroups. If S i s a Sylow system of a group G we can define the system normaliser, N = N(S), to be a l l those g i n G that transform each member r of S i n t o i t s e l f . A l t e r n a t i v e l y we can define N by, N = 0 Ng(S^). i = l Of some i n t e r e s t i s the cl a s s W defined by : W = { G | the system normalisers of G are s e l f normalising}. The groups i n W are c a l l e d S-C groups. Huppert [ 6 ] , gives many of the known properties of system normalisers and S-C groups. Carter [1], proved the existence of s e l f normalising n i l p o t e n t subgroups i n any group and showed these were a l l conjugate. For the next r e s u l t we need two standard r e s u l t s about the-system normalisers of a soluble group, namely ; (1) Every system normaliser covers each c e n t r a l chief f a c t o r , and (2) Every Carter subgroup contains a system normaliser. 3 • • • . (1.2.9) Theorem : [Alperin Thompson] S(W) = -S , Proof Huppert [7]. Let G e 3 and 1 = G 0<d G - L 4 * ' " 4 G N =' G be a chief s e r i e s f o r G. Let = G^/G^_^ i = 1, • • *m and |K^| = p ^ 1 . Choose a prime p d i f f e r e n t from a l l the p / s and choose n^ so that k ± <_ n ± and p " 1 = l( p ) . Let N ± = G F ( p n ± ) + . (a) By a simple induction, we show G can be embedded i n the i t e r a t e d regular wreath product ( • • • (N-^  wr. N2)**') wr. N m . As G -—> Gm_2_ wr. G/G J J J.! > [ ( • • • ( % wr. N 2 ) " « ) wr. N ^ ] wr. N m. (b) Every has an automorphism of order p that f i x e s no non t r i v i a l elements ; namely that induced by l e f t m u l t i p l i c a t i o n by a pr i m i t i v e p root of unity i n GF(p x ) . Claim : The group "H^. = (•••(Nj wr. N2)''*) wr. N^ . has an elementary • Abelian group of automorphisms U|c with [u^ .[ = p k , and such that no 1 ^ h e H K i s f i x e d b y . a l l a i n U^ . For k = 1 set U-^  = <a > . Suppose that claim i s true f o r and consider the group H ] £ + ; L = B^ . wr. N ^ - j . -By (1.1.1) there corresponds to the automorphism y = ot^ +j of a n automorphism y of order p of H^+i defined by ( f , n ) y = (f * , n y ) where f'(n) = f ( n y _ 1 ) . The i d e n t i t y , ( for s u i t a b l e g and 1 f n e N^ +^), ( f , n ) y ( f , n ) _ 1 = (g.n^n - 1) shows the automorphism y acti n g on H^^/H^ wr.l, f i x e s no element. By (1.1.1) we can define the automorphism cT e Aut(H^ +^) corresponding to a e U k by ( f , n ) a = ( f ' , n ) , where f*(x) = f ( x ) a . Then \={a | a e Uk> i s a group of automorphisms of Hk+-^ isomorphic to . Let Ujc+-^=<U|C>[I> . By our induction hypothesis i s elementary Abelian of order p k and by d e f i n i t i o n y and a e U^ . commute. So H k +^ i s elementary Abelian of order p k + l . Also i f ( f , n ) T f o r a l l T i n Uk+-[_, then by considering T = y" and T = a E Ujj , we see that f = n = 1 . Thus the claim i s proved. (c) Let H = HjjjlLjj the semi d i r e c t product of by U m . We have chosen p such that ( | H j , | u m | ) = 1 and so can apply Huppert [ 8 ] . Let x e HJJ, H N H ( U m ) . Then as H ^ H, for a l l a i n U m, [x,a] e PL^ f\ U m = 1 . So x e C H(U ) n H m = 1 . Hence N H(U m) = U m and U m being Abelian i s thus a Carter subgroup of H. A system normaliser of H covers a l l c e n t r a l c h i e f f a c t o r s and thus covers U m, as H/IL^ = U m i s Abelian. Also a system normaliser i s contained i n each Carter subgroup. Thus U m i s a system normaliser f o r H which i s s e l f normalising making H a S-C-group. By (a) G i s isomorphic to a subgroup of H, completing the proof. A group i s c a l l e d p r i m i t i v e i f i t has a f a i t h f u l p r i m i t i v e permutation representation. This i s equivalent to saying the group has a complemented unique minimal normal subgroup. It turns out that t h i s subgroup i s the F i t t i n g subgroup. Thus another c h a r a c t e r i s a t i o n of p r i m i t i v e groups i s those that possess a s e l f c e n t r a l i s i n g minimal normal subgroup. We c a l l the c l a s s of p r i m i t i v e groups P. A subclass of P, i s defined by PQ = { G | Q(G) <_ P } . A l t e r n a t i v e l y can be defined as the class of groups with a unique complemented c h i e f s e r i e s . This i s also equivalent to s p e c i f y i n g Suppose P i s a p-group i n pQ. Then as P i s i n P, we have <j)(P) = 1. Hence F(P)/<|>(P) = P, i s an elementary Abelian p-group. But the only elementary Abelian p-groups i n P^ are those of order q, q a prime. Let X be the clas s of groups i n which every chief f a c t o r i s complemented. Then we have the following Lemma. (1.2.10) Lemma : If G e X and G , then N i s complemented i n G . Proof : We use induction on |G|. Let H-4 G and N ° ^ G , such that N <_ H. In (2.1.1) we show QX = X, so G/N e X and | G / N | < | G | . As H/N< j^G/N, by the induction hypothesis, H/N i s complemented i n G/N. i . e . there e x i s t s L/N such that L/N • H/N = G/N and L/N H H/N = N/N. The chief f a c t o r N has a complement M i n G . We consider M O L. By Dedekind's modular law, N(M H L) = 1 fl L = L, and N n (M (\ L) = 1. Also H (M n L) = HN(M H L) = HL = G ; and (M n L) fl H = M H N = 1. So M fi L i s a complement of H i n G . C l e a r l y P^ < X, so i f E<§ G e pQ, then N i s complemented i n G . I t i s of i n t e r e s t to know what the subgroup closure of pQ i s . (1.2.11) Theorem Pardoe [1]. S(pQ) =5 Proof : We w i l l give a survey of the method. Let G E S. We w i l l use induction on |G| . Let L<3">G and G/L = Gy. By induction there i s a monomorphism, u : L — > M, where M e P ^ . As the ch i e f factors of groups i n pQ are s e l f c e n t r a l i s i n g , H'-<30M and M/MT = C^ . We choose a prime p ^ q or r. Let = M, i = 1, 2, p, and l e t W = M wr. C p = (Mxx. • . x M p ) C p . I f NJ; = M| , i = l , - - - , p , and N = N j X . - . x N p , then W = W/N = C q wr. C p . Let S be a minimal normal e c c e n t r i c subgroup of W contained i n the base group. Let S be the inverse image of S under the natural homomorphism : W — > W. Define T = SC p. We w i l l c a l l the construction of T, "Pardoes Construction", and w i l l r e f e r to i t as such l a t e r on i n t h i s proof and i n Chapter '2. I t i s then shown that, (1) L can be embedded i n S and, (2) T E P Q , provided pjq - 1. The proof of (1) i s routine and w i l l be omitted. The proof of (2) contains the crux of the argument. Now T = SCp i s c l e a r l y i n P ^ . We claim that any ch i e f s e r i e s of G must pass through N. For l e t , \ = { ( l , . . . , h ± , . - - , l ) | h ± e Ht} £M - [ X...xM p , where M i . Then, _ a S/CgCH^) = M i/C M (Hj.) f o r a l l i = 1, - . - . p . where a : 'sCgCH^ = ( s l s • • •, s±, • • •, s p ) C g (IL) —> s ^ O L ) . i s the required isomorphism. Let K«£2,T such that, K N and N ^ K. The following diagram of normal subgroups i s then a p p l i c a b l e . KN K N As N < KN<3 T, then K N >_ S . Also [ K , N ] <_ K f) N , so K / ( K H N ) c e n t r a l i s e s N / ( K H N ) . Thus K c e n t r a l i s e s a l l chief factors of T between K H N and N . Let N / H be a chief f a c t o r of T. As N / H i s Abelian C T ( N / H ) >_ N and combining these two r e s u l t s , C T ( N | H ) ^ N K > _ S . Let H^ -<1 N.<lM. be the top por t i o n of the chief s e r i e s f o r M^. Let N/F = ^ x • • • x N i N^ x ... x H. x N From what follows below, H can be P H ± chosen as x... x R_, so N/F e Q(N/H) and hence C,r(N/F) >_ C T(N/H) _> S. Then 1 = S/Cg N. H i . M i/C M_ L(N i/H i) = — i 1 . For, M ± e P \ g i v i n g Cj^_(N^/H.) = N.. Thus no such K can e x i s t . This concludes the proof that i i x i f K <J T then e i t h e r K <_ N or N <_ K. Next we show i f , 1 = M. , < M. . _<3 i , k i , k - l M. „ <3 N. <! M. i ,2 i i i s the chief s e r i e s f o r M^, then M_. . x ... x M ./M. ... x  •• ^  i , J p ,j i,3+1 p »j+l i s a chief f a c t o r of T , when j >_ 2. For, by induction on |G| we can assume that the chief s e r i e s f o r i s , 1<1 A<3 N^<3 M^, and i t i s s u f f i c i e n t to show A x MA^ x x A p o<J T. This i s done by noting that, i f U » 4 T and U <_ A, then A^ <_ U f o r some i and hence A <_ U. The e s s e n t i a l point i n the argument comes i n showing that N o < : l T, where we have assumed i n d u c t i v e l y that N.«<1 M. . If £| | N ^ | , £ a prime, then Z f q as the only q groups i n PQ are c y c l i c of prime order. I f we consider N as a S-module under conjugation, then the are i r r e d u c i b l e S-submodules and S/N = S i s an elementary Abelian q-group. Let S = NQ where Q i s a Sylow q-subgroup of S. Choose a p such that pj q - 1 . Then S i s a f a i t h f u l i r r e d u c i b l e module f o r C„ and by ( 1 . 1 . 3 ) dim 7 (S)= P \ = smallest integer n, such that p l q 1 1 - 1 . Hence dim„ (S) > 1 , and as — ~ 2 S = Q i s an elementary Abelian q-group of order at l e a s t q , Q i s not c y c l i c . Consider N as a Q-module. By Maschke's theorem N i s a semisimple Q-module and, N = N-L © N 2 • • • $ N p , i s the decomposition of N i n t o i r r e d u c i b l e Q-submodules. Let = C Q ( N ^ ) . Then N ± i s a f a i t h f u l i r r e d u c i b l e Z ( Q / C Q ( N 1 ) ) module and by ( 1 . 1 . 3 ) , Q/Cq(Ni) i s c y c l i c . Thus H ± f 1 . Also U± f Q by the remark that S/CgCH^ = M ^ / C ^ O L ) . • Suppose = Nj for some i ^ j . Then = . The a c t i o n of C p i s to permute the N^, so there i s a g i n C p, such that = Nj and C p = <g> . But "S = QN/N O f so, Q s = Qg/N H Q§ = Q/N H Q = Q . Hence C Q ( N 1 ) = ( CQ(N£))§ and N C Q ^ X J T, c o n t r a d i c t i n g the f a c t that "s«< T . So the N. are non isomorphic i n p a i r s and hence the only submodules for N are d i r e c t sums of the N^s , i =].,•••,p.. Thus any non t r i v i a l submodule contains a N^ and by the a c t i o n of C p permuting the N^, contains N. So N « < ] T . To show T i s i n i t i s s u f f i c i e n t to show that the chief s e r i e s 1 <^ J • • • <3 M-^  2 x x ^ p 2 < ^ N < ^ S < ^ - ' - ^ s complemented and each chief f a c t o r i s s e l f c e n t r a l i s i n g . These are routine and w i l l be omitted. The f i n a l step i s broken down into two cases. (1) Suppose r f 2 . We repeat the above construction with r instead of f? and T instead of M. Now T' = S i s the maximal normal subgroup of T and p replaces q. By (1.1.6) there are an i n f i n i t e number of primes p such that p = 2 + nr, (some n), so i t i s possible to choose a prime p such that pjq - 1 and r j p - 1. The new group U obtained contains S as a subgroup and by (1.1.2) G > L wr. G/L > S wr. C r > U <_ T wr. C r So G can be- embedded i n the group U e P ^ . (2) Suppose r = 2. Since 2|p - 1 f o r a l l odd primes p, at this stage we w i l l have to resort to a d i f f e r e n t method of proving that "N°<lT". Without los s of g e n e r a l i t y we can assume ' 5 { q - 1 and apply Pardoe's construction to M wr. C^. Let X be the group i n P ^ so formed. For, suppose 5|q - 1. Then by (1.1.6) there i s a prime s such that s j q - 1 and 5 j s - 1. Then we would apply Pardoe's construction to M wr. Cg to form a group A e P^  . Then we would apply Pardoe's construction to A wr. C5 to form B e P^  . We would then complete the proof with B instead of X. Apply Pardoe's construction to X wr. and note that 31J5-1 and -3115"^—1 . I f Y i s the group i n P^  so formed apply Pardoe's construction to Y wr. C 2 to form the group Z. These constructions give us a s e r i e s of groups with normal s e r i e s . (a) T C c (b) 31 c3 Y U (c) 31 c6 I I Z V UxU=R S 1 R e c a l l that for the chief s e r i e s below S we j u s t use the method of the theorem, as the fact that p | q - 1 was only used to show "N°<3 T". Let R = R/S and "z = Z/S. Then i t i s s u f f i c i e n t to show R » < Z . We can consider R as a vector space of dimension 6 over Z^ and a D = Z/R-module by conjugation. Note R considered as a A = V/R , ( = C ^ ) , module i s the d i r e c t sum of two isomorphic i r r e d u c i b l e modules X and Y, of order 5"^. So R] = X # Y Suppose R i s not an i r r e d u c i b l e D-module. The only possible i r r e d u c i b l e submodules are X or Y. As A<J D, A i s Abelian and D e P Q, then C D(A) = A. Thus by (1.1.5) C 2 = D/C D(A) i s isomorphic to a subgroup of Cg. This i s impossible and thus R i s an i r r e d u c i b l e module f o r D. Thus R<»<3 Z . This completes the proof that Z and hence Z belongs to P^. To complete (2) we note that there are monomorphisms G >- L wr. G/L > S wr. C r > Z which completes the proof of the theorem. §2.1 : P r e l i m i n a r i e s . In t h i s s ection we w i l l i n v e s t i g a t e the r e l a t i o n s h i p between and the l a r g e r c l a s s of groups X defined by X = {GJ every chief f a c t o r of G i s complemented} . Carter, Fischer, and Hawkes [1] have shown that i f a group G has a chief s e r i e s a l l of whose factors are complemented then every chief f a c t o r i s complemented. Thus we may define X = {G| G has a chief s e r i e s i n which the c h i e f factors are complemented} . I f y i s the cla s s of groups with t r i v i a l F r a t t i n i subgroup then X also may be characterised by X = { G | Q(G) £ V } =" { G j For a l l K<J G, <J»(G/K) = 1 } Theorem (1.2.11) shows that the class P^ i s large i n the sense that every soluble group can be embedded i n a P^ group. What can be said about the groups that can be subnormally embedded i n a PQ group ? As pQ <_ X we have SnP°- <_ SnX = X. However we s h a l l see that SnP°- < X . We are l e d to ask whether there i s a natural c o l l e c t i o n of closure operations whose j o i n applied to enlarges i t to X. C l e a r l y X i t s e l f must be closed under each of the operations appearing i n the c o l l e c t i o n . We f i r s t examine these. ( 2 . 1 . 1 ) Lemma : X = {Q, S n RQ, D Q}X . Proof : ( i ) From the a l t e r n a t i v e d e f i n i t i o n of X c l e a r l y QX = X. ( i i ) Let H<3 G e X. Suppose 1 = HQ < H x < < H r = H<! G be part of a chief s e r i e s of G running through H. By C l i f f o r d ' s Theorem H^|lL_^ i s completely reducible as a H-module. I f L complements H i / H ^ _ ^ i n G then L fV H complements H-j/H ^ i n H. I f N / H ^ o ^ H/H.^ , with N < H., then H-/H. , = N/H. .. x NA/H. .. with N* < H. So that — i ' i l - l . l - l i - l — (L H H)N* i s a complement of N/IL ^  i n H. Thus H has a chief s e r i e s i n which every chief f a c t o r i s complemented. Hence H e X and SnX = X. ( i i i ) Suppose G i s a group of minimal order i n RQX - X . Then there e x i s t N x, N2<3 G , % H N 2 = 1 , and G / % , G/N 2 £ X . We may assume N-L«><} G . For, l e t N 0 < d G such that N <_ Nj_ : Now consider G/N. It has normal subgroups N-^ /N and N2N/N which have t r i v i a l i n t e r s e c t i o n . Also .G/N^ N-j/N = Gl^1 e X and G/N/N2N/N = G/N ?_/N 2N/N 2£ QX = X. Thus G/N e RQX and as |G/N[ < | G | i t Is also i n X. As G/N ± E X then <f> (G/N-±) = 1 , i = 1 , 2 ; and <|>(G) <_ N x f\ N 2 = 1 . Now G/N-^ as a member of X has a chief s e r i e s i n which every chief f a c t o r i n which every chief f a c t o r i s complemented. Thus G e X and RQX = X. (iv) I f H and K are i n X and H K are respective chief s e r i e s f o r H and K i n which each chief f a c t o r i s complemented then, 1 = H Q <f E±<i • • • <1 H r < J H r x K]_ <3 • • • < q H r x K s = H x K .is a chief s e r i e s for H x K i n which each chief f a c t o r i s complemented. The case f o r the d i r e c t product of a f i n i t e number of groups i n X follows by a simple inductive step and hence DQX = X . The c l a s s X i s not S - or N D - closed. The symmetric group of degree 4 i s i n X but i t contains a subgroup isomorphic to the di h e d r a l group D of order 8 ; a group which has a non t r i v i a l F r a t t i n i subgroup. Also D i s the normal product of two K l e i n 4-groups and so D e N 0X _ X . Consider the semidirect product G of Cy by i t s automorphism group Cg. Then G i s the product of two normal subgroups, N-j_ = Cy«»C2 and N 2 = Cy"C^ which are i n ' and which have coprime index i n G. In §2.3 we show G e X . This group G o f f e r s on i n s i g h t into the i n v e s t i g a t i o n of a closure operation less general then N Q and under which X may be closed. 1 = H Q< H]_ 4 • • • -4 H r = 1 = K Q < 1 • • • < Ks = §2.2 Two Closure Operations . The example of the holomorph of C^  suggests we might define an operation T on group classes by TV = (G | G = N1N2; N1,N2< G; N l 5N 2 e X ; ( |G : Nx|, |G : N2| ) = 1} • This definition, however, does not give us a closure operation as i t is not idempotent. But recall the alternative method of defining a closure operation in §1.1. It is sufficient to specify a l l the T-closed classes and that the collection C of T-closed classes satisfy ; (1) S e C ; (2) C is •closed under taking arbitary intersections. We will say the class V is T-closed i f and only i f a l l groups G, which are the normal product of two V-groups of coprime index in G, are in y. Clearly the T-closed classes satisfy the conditions (1) and (2), and so associated with this collection of T-closed classes is a closure operation which we will denote by T. This description is in fact identical to one given by Kappe [1] when we restrict our attention to the universal class S. The next question to ask is what is the action of T on an arbitary class of groups V. This question has not yet been answered but its investigation led to the description of a new closure operation which we call M. The definition of the operation given at the start of this paragraph failed to be a closure operation because i t was not idempotent. To overcome this we define M by; nY = { G I G = <X p • • • ,Xn> ; X ± s n G ; X ± e V ; ( | G : X-J,- | G : x j ) =• 1 '.} This operation i s c l e a r l y expanding and monotonic. I t remains to show i t i s idempotent and hence a closure operation. I f G e M(MY) then G = < Y L 5 • • • Y N > where the Y . ^ are subnormal i n G and belong to M^. Also i f we l e t y^ = | G : Y ^ | then ( y ^ , • • • , y n ) = l . Each Y . = <X.,,,,i,X. > where the X.. are subnormal i n Y . and belong to y. Also i f x ± j = | Y ± : X ^ |. then ( x ^ , • • • , x i r ) = 1. But the X ^ are i n X and subnormal i n G and i f x ^ = | G : X ^ | , we claim that ( x 1 1 , ' ' ' , x n r ) = 1. For suppose p | x.. for a l l relevent i and j . Then p| | G : X i j j = | G : Y ^ | | Y ^ : X^j | = y ^ x ^ j * Bor a given i there i s at l e a s t one x.. not d i v i s i b l e by p, so p|y. for a l l i . This c o n t r a d i c t i o n proves our contention and M i s idempotent. P h i l i p H a l l [2] has remarked that i f H-^ , ^ j * * * , H r are subgroups of a soluble group G such that ( | G : H^|, •••». |G : H r| ) = 1 then G = H^H2'**Hr . Let P-^ , P2,**,,PS be a representative set of sylow p^ subgroups of G f o r each d i s t i n c t prime P J _ | | G | , i = l , , , - , s . Then there i s f o r each i a j such that P^ <_ H^  . Thus the remark i s equivalent to saying that P i - ^ " * *^s = G» where the ordering of the product i s not s p e c i f i e d . This follows by induction on |G| as G being soluble means one of the P^ contains a proper normal subgroup. Thus i n the d e f i n i t i o n of M we may say G i s the product of subnormal subgroups and not merely generated by subnormal subgroups. The closure operation M i s s t r i c t l y less than N Q as the dihedral group of order 8 i s i n NQA but not i n MA . We can generalise M to a family of closure operations , where IT i s a set of primes, by de f i n i n g ; MY =.{G | G = X 1X 2--'X n ; X ± sn G ; X ± e V ; ( |G : X - L j , • - -, | G : X n| ) i s a Tr-number} ^ when TT i s the empty set we get M and when TT = { a l l primes} we get N Q. It i s c l e a r from the d e f i n i t i o n that T <_ M, and i n fac t t h i s i n e q u a l i t y i s s t r i c t . For consider ; (2.2.1) Example. Let A and B be groups i n P^ formed by Pardoe's construction (1.2.11) from C 2 wr. Cg and C7 wr. C5. That i s A has a • 2 unique minimal normal subgroup of order 2 with f a c t o r group c y c l i c of order 3 and B has a unique minimal normal subgroup of order 7^  with f a c t o r group c y c l i c of order 5. Let C be a subgroup of A of order 2 and l e t D be a subgroup of B of order 7. Let N-^  = A x D and N 2 = C x B be subgroups of G = A x B, and set V ='•{!, N-j_, N 2) . Then G e ViV - / . On the other hand V i s T-closed. For i f K e TV then K i s the normal product of two members of V of coprime index i n K. We may suppose that neither of these i s 1 as i n th i s case K = Nj_, i = 1,2, and K e y . Then K = NjjN2 = A x B = G. But and N 2 are not normal i n G so t h i s case i s excluded. But t h i s has exhausted a l l the p o s s i b i l i t i e s f o r TV and our claim stands true. Hence T < M . (2.2.2) Lemma : I f Y = SnV and G e generating normal i n G. then we may choose the Proof : Let G = '^n where the X^ are i n Y, subnormal i n G, and i f |G : X^ | = then (x-^, • • •, x 1 1) = 1. Let cr(x^) = {p| p| x^} and T N = o(x:j_)' , the complementary set of primes. Let H^ be a H a l l subgroup of x^. Then by the d e f i n i t i o n of T T ^ , H^ i s a H a l l T T ^ subgroup of G. i . e . H. = G . I t i s a w e l l known f a c t that i f a H a l l Ti-subgroup i s contained i n a subnormal subgroup then so i s the j o i n of a l l i t s conjugates. Hence <G^> £ X . . Thus > <_ Core X ± and a( |G : Core X i| ) = a (xj_). Thus i f x^ = |G : Core X^|, we have (x-|_, • • • ,x n) = 1 . We now show G = = Core X]_. Core X 2- ••• Core X n . Let p i , i = 1,2, r, be the d i s t i n c t prime d i v i s o r s of |G : G'| . Then there i s an x^ such that p. | . Q If G„ i s a Sylow P^-subgroup of G, then P. = <G p > > <_ x.^  . by the same i 1 i G reasoning that < G 7 r < > f_ x±- Then P^<3 G and P^ <_ Core X^. We claim P = P 1 P 2 " - P r = G. If not, then P < G and hence there i s an M<3 «G such that P <_ M. I f (G : M) = q, then q = pj ; some j <_ r, since G' <_M. But <Gq> = P^ <_ P <_ M and therefore q j |G : M|. This i s a cont r a d i c t i o n and hence P = G. Thus G = <Core X i| i = l,---,r> and as V •= S n/, the subgroups Core X^ are i n V, normal i n G, and s a t i s f y (x-^, • • • ,~xn) = 1. This complete, the proof. (2.2.3) C o r o l l a r y : I f V = {S n, Q}V then 0}lY = Proof : I f H e Q K Y , then H = G / K where K <3 G and G e Ml'. As V = SNY we can assume G = X-^ X 2 ' " ' X N , where the X ^ are normal i n G , belong to Y, and ( x ^ - - - ^ ) = 1 . In G / K we have X ^ / K ^ G / K and X ± K / K i n V as OY = V. Also | G / K : X±K/K\ | G : X ± | , so i f x± = = | G / K : X ± K / K [ , then ( x l 5 x 2 , • • • , x n ) = 1 . Thus G / K e KY. § 2 . 3 . A Further i n v e s t i g a t i o n of X. The holomorph of C-j was the motivation f o r i n v e s t i g a t i n g the closure operation M. At the end of § 2 . 1 we stated i t was i n X. We now prove that claim. ( 2 . 3 . . 1 ) Lemma : MX = X .. Proof : We have shown SnX = X so by ( 2 . 2 . 3 ) we may assume ; i f G e MX, then, G = X]X2"'xn , X ± e X, X ±<f G, and i f x ± = |G. : X I | then (x]_, • • - ,x n) = 1 . Let G be a group of minimum order i n MX -X. Let N«<3 G and as |G /N | < | G | and QX = X by the induction hypothesis and ( 2 . 2 . 3 ) then G / N e X and has a complemented c h i e f s e r i e s . If N = P01, there i s an x^ such that pjx^ and thus X ^ contains a l l the p~subgroups of G. Hence N < _ X ± . Then ( | N | , | G : X ± j ) = 1 and N i s an Abelian normal subgroup of G contained i n X ^ . But ( 1 . 2 . 1 0 ) shows that every normal subgroup of an X group G i s complemented i n G. X ^ e X implies that N i s complemented i n X ^ and therefore by Gaschutz's theorem [ 1 ] , N i s complemented i n G. So G has a complemented chief s e r i e s and thus MX = X. We w i l l now inves t i g a t e the ac t i o n on PQ of the j o i n s of some of those closure operations that leave X f i x e d . But f i r s t we w i l l introduce a us e f u l normal s e r i e s . The Upper F i t t i n g Series of a group G, 1 = F D(G)<! F1(G)<1 . . . <3 F £(G) = G i s defined i n d u c t i v e l y by F D(G) = 1 and F ±(G) / F ^ C G ) = V(G/F±_±(G)). We c a l l the length £(G) = £ of t h i s s e r i e s the F i t t i n g length of G. For a group i n P^ the upper F i t t i n g s e r i e s coincides with the unique chief s e r i e s . (2.3.2) Lemma : If H sn G e PQ then F i ( H ) / F i - l ( H ) E S ( F i ( G ) / F i - l ( G ) ) ' i = 1,-",A(G>. Proof : The proof follows by induction on £(G). I f l(G) = 1 the re s u l t i s c l e a r l y true. So suppose the hypothesis i s true f o r a l l groups K such that £(K) = I - 1. Let l(G) = I. Then £(G/F 1(G)) =1-1 and G/F-j^G) e QX = X. Also i f H sn G then 11F 1(G)/F 1(G) sn G/F 1(G). But as F X(H) = F X(G) fV H we have HF 1(G)/F 1(G) = H/F-jiOl). Thus by the induction hypothesis F j L(H/F 1(H))/F i_ 1(H/F 1(H)) • e S [F ±(G/F 1(G)) / F i _ 1 (G/F][(G) ) ] for i = - 1. But for any group K , F i ( K / F ^ K ) ) / F i _ 1 ( K / F 1 ( K ) ) = F i + 1 ( K ) / F ± ( K ) and the lemma i s s a t i s f i e d f o r i = 2,''-,Z . The case i = 1 i s covered by the remark that F^B.) = H H F^(G). (2.3.3) C o r o l l a r y : I f -H i s an Abelian subnormal subgroup of a group i n pQ then H i s an elementary Abelian p-group f o r some prime p. Proof : I f H sn G e PQ and H Abelian then F 1(H). = H. By (2.3.2) F^(H) <_ F-^(G) and F-^(G) i s elementary Abelian for some prime p as i t i s the unique minimal normal subgroup of G. (2.3.4) Example : Sn(P )^ < X . The Abelian group Cp. x C3 i s i n X but by (2.3.3) i t cannot be a subnormal subgroup of a group i n P^  . We know X i s R Q - and S n~closed, and, i n f a c t , we can show R Q S n i s a closure operation. For, i f V i s a clas s of groups and G e S nR oy, then G sn H where H has r normal subgroups H.^  with t r i v i a l i n t e r s e c t i o n such that each H/H^ i s i n V . Consider the r normal subgroups H-^  H G of G. They have t r i v i a l i n t e r s e c t i o n and G/'(HinG) = GH i/H ± sn H/H± e V . Thus G i s i n R 0S n^ and S nR Q <_ R Q S n . This i s a s u f f i c i e n t condition f o r R Q S n to be a closure operation. (2.3.5) Example : R Q S n(pQ) < X . F i r s t observe that, i f K i s a group i n P H R0^, then K i s i n V. This follows from the d e f i n i t i o n of P and R Q-closure. Consider the bolomorph G of Cy. Then G i s i n P (\ X and the upper F i t t i n g s e r i e s for G i s l<QCy<3 G. Thus G i s not i n S n(pQ) by (2.3.2) as the factors i n i t s upper F i t t i n g s e r i e s are not a l l elementary Abelian p-groups for various primes p. Hence G i s i n X but, by the i n i t i a l observation, not i n R DS n(pQ). Another l i n e of attack could be to consider the closure operation D QS n applied to P^ . But t h i s i s not s u f f i c i e n t as shown i n the following example. (2.3.6) Example : D 0 S n ^ ^ ^ < * • L e t G b e t h e s e m i d i r e c t product of Cg x by C 2, where the a c t i o n of the generator x of C 2 i s to transform (a,b) i n Cg x to (a~"'~,b_-'-). Then G i s i n X and i s not the d i r e c t product of any two proper subgroups. So, i f G were i n D 0S n(P^), i t would be i n S n(pQ), which by (2.3.2) i s not p o s s i b l e . Another l i n e of i n v e s t i g a t i o n i s the closure.operation QS n applied to pQ. To decide whether X <_ QSn(P°0 we can begin by showing c e r t a i n subclasses of X are i n QS n(pQ). One such subclass i s X (1 A. C l e a r l y the simple Abelian groups are i n X, and as DQX = X, so are the elementary abelian groups. But these are a l l . For i f A i s an Abelian group, then F(A)/cj>(A) = A/<J)(A) i s elementary Abelian. Hence, i f A i s not elementary Abelian, then <j>(A) i s non t r i v i a l and A i s not i n X. In what follows we w i l l make use of the following non standard notation. I f G i s a group and n a p o s i t i v e integer define, nG = = G1 x G 2 x x G n , where G ± = G., i = 1,2,•••,n. Let K be a group i n and suppose |K| = p, where K = K/K'• Let q be a prime such that qjp - 1. Consider A = K wr. Cq = C p wr. Cq. If B = qK and ~B = qK/qK', then A = BC q . By Pardoes construction (1.2.11) we can form T = L C q i n pQ where L = L/qK' i s a minimal normal e c c e n t r i c subgroup of A contained i n B. Let b be the smallest p o s i t i v e integer such that q [ p^ * - 1. By (1.1.3), |L| = p^. Since qK'<l T e P^ , by (1.2.10), i t has a complement H i n T. Let L* be the complement of qK' i n L. i . e . L* = H H L. We note that L* = L , and the following holds. (2.3.7) Lemma : There e x i s t s a subgroup W* of L* such that (a) |W*| = p 1 3- 1 . (b) [K q, W*] = 1. Proof : Consider BCq = Cp wr. Cq . B i s an elementary Abelian p-group of order p^. We can chara c t e r i s e B as a q-dimensional vector space V over the f i e l d of p-elements. For i f {u^,'-',Uq} i s a s u i t a b l e basis f o r V and C p = <x^| x? = 1>, i = 1,•••,q, then the isomorphism i s given by, a l a2 v a : ( x 2 , • • • ,x ) > I a ± u i ; 1 <_ a ± <_ p, i = 1,2, • • • ,q . q i = l Let 3 be the map, q I 1=1 3 : I a.u. > a q . R e s t r i c t the map a to L - L and consider the map 3a on L*. The a l a q kernel of t h i s map i s {(x^ )| a q = p} and dim a (L*) = dim ga(L*) + dim (ker ga) Thus there i s a subspace of dimension b - 1 of a(L*) i n which a l l elements have a q = p (=0). When considering L* as a subgroup of L t h i s means there i s a subgroup W" of L* of order p^ \ i n which every element i s of the form ( y ^ ' - y ^ , ! ) • Now K q = { ( 1 , k q ) | k q £ K q} , so c l e a r l y [W*,K q]=l. (2.3.8) Theorem : A n X <_ QS n(pQ) . Proof : Let nCp = Cp x ••• x Cp, (n times). Then i t i s s u f f i c i e n t to show m-jCp x x mrCp i s i n QS n(pQ) where PpP2» * * * »P r a r e d i s t i n c t primes. When do t h i s by induction on r . When r = 1, consider mCp. Choose a prime q > p and look at Cp wr. C q = BC q. Let A be a minimal normal e c c e n t r i c subgroup of BC q contained i n B. Form W = AC q . Then W i s i n pQ as i t has a unique complemented chief s e r i e s , 1<JA<3W. Let a be the smallest I a p - 1. Then a > m by the choice of q. Also A = aC p by (1.1.3), and mCp sn aC p <l.W. So mCp e Sn(P°0 . Thus we may assume QS n(pQ) contains a l l elementary A.belian groups with r - l d i s t i n c t prime d i v i s o r s . Now consider the case of r d i s t i n c t prime d i v i s o r s p^»P2»* *'>Pr- We can order these such that P r^2. Let G = m^Cp^ x ••• x m rC p and G ^ = m l c p 1 * *'' x m r - l C P r _ i * Let G-L = A/B where, B <3 A sn K e PQ . We can always choose A, (or K ) , such that A <_ K ' . For, i f A K ' , then A = K as K ' i s the unique maximal normal subgroup of K and a subnormal subgroup passes through a normal one. In t h i s case G-^ e and thus must be c y c l i c of prime order. Then as i n the case f o r r = 1 a K can be choosen such that A <_ K ! Let K / K ' = K be a c y c l i c group of order p f o r some prime p. CO As p T T 2 we can use D i r i c h l e t ' s r e s u l t that the sequence {2 + np„} .. 1 r r n=I contains an i n f i n i t e number of primes r. Thus i t i s p o s s i b l e to choose a prime r such that p ^ r - 1 and r-fp - 1. Consider K wr. C r = C C r = C p wr. C r. As r^p - 1 we can use Pardoe's construction to form a group, L = R C r e pQ, where R i s a minimal normal e c c e n t r i c subgroup of CC r contained i n C. Let L = L/L' and consider L wr. = DC^ = C r wr. Cn . Again, by Pardoe's construction we can form a group H = SC p where S i s a minimal normal e c c e n t r i c r subgroup of DCp contained i n D. Since p |r - 1, we have H. e pQ. F r r m +1 _ Let s be a prime such that s > p^ . Let H = H/H', and consider H wr. C s = EC S = C p wr. C s. By Pardoe's construction once more we can choose T a minimal normal ecce n t r i c subgroup of EC S contained i n E and form J = TC S . As s \ p r - 1 then J e P ^ . We claim G s Q S N ( J ) . The group J has been constructed i n the following steps. T J < J ' c i L' 4 P, q? K - B J=TC S , s J'=T sH1 bs sp L' asp rsp rK' By (1.1.3) the integers a,b,c are the smallest such that r | p a - 1, p r | r b - 1, s|p£ - 1, r e s p e c t i v e l y . Let SH' = R| x ... x H g , where ILj = H' . Now there exists a subgroup U of T that complements SH' which, by (2.3.7), has a subgroup V, such that |v| = p £ - 1 and [V,Hg] = 1. By the choice of s, c > mr, so we may choose a subgroup m W of V such that |w| = p r r and [W,HS] = 1. Consider the subgroup (Hj_ x ... x H g_ 1 x A).W where A sn K' = KjL . Then, (H^ x • • • x Hg_1 x A). W sn (H| x ... x Hs_^ x Kl)W sn (Hj_ x . .. x H l ^ x HI)W sn J Also (ILj x ... x Hg_1 x B) <J (Hj_ x ... x x A)W as B <3 A and centralises Hs . Finally, (Hi x x IVS_± x A)W a _ H^  x ... x H^_1 x B = m ^ x ... x m r C P r . where a is given by, a[(h1,«'-,hs_1,a)w.Hj_ x ... x Hs_^ * B] = (aB, wH[ x . • • x Hs) . This completes the proof. CONCLUSION We conclude with some unanswered questions that follow from the above Chapter. First is QSnPQ = X ? It would seem that the answer to this is no. For, let L be the class of groups whose upper Fitting factors are p-groups for various primes p>and V be the class of groups whose chief factors are L-central i.e. satisfy G/Cg(H/K) e L for a l l chief factors H / K of G. If we can show V = QSnV then we are done. For pQ <_ V and the holomorph of Cy is contained in X — V . Let K<3 G e y and H / K / L / K be a chief factor of G/K. Then H / L is a chief factor of G and as C G / K ( H / K / L / K ) = C G ( H / L ) / K we have QV = V. Thus we need only show SnV = y to complete the proof of the existence of the counter example. This is at the moment unsettled. Finally, given that the first question is settled, we may then ask is (Q, Sn, M, R0}PQ = X ? BIBLIOGRAPHY 1. A l p e r i n J.L. : [1] The construction and c h a r a c t e r i s a t i o n of Some Classes of F i n i t e Groups. Archiv. der Math. 15, 349-354 (1967). 2. Blessenohl and Gaschlitz : [1] Uber Normale Schunkund F i t t i n g K l a s s e n . Math. Z. 118, 1-8 (19 70). 3. Carter R.W. [1] Nilpotent Self Normalising Subgroups of Soluble Groups. Math. Z. 75, 136-139 (1961). 4. Carter, Fischer, Hawkes : [1] Extreme Classes of F i n i t e Soluble Groups. J . Algebra 9, 285-313 (1968). Lemma (2.6). 5. D i r i c h l e t G.C. [1] French T r a n s l a t i o n . Jour, de Math. 9, 245-269 (1844). 6. Fischer, Gaschutz and Hartley [1] Injektoren Endlicher Auflosbarer Gruppen. Math. Z. 102, 337-339 (1967). 7. Gaschutz W : [1] Zur Erweiterungstheorie Endlicher Gruppen. J . Math. ' 190, 93-107 (1952). 8. H a l l P. [1] A Note on Soluble Groups. J.L.MS. 3, 98-105 (1928). [2] A C h a r a c t e r i s t i c Property of Soluble Groups J.L.M.S. 12, 188-200 (1937). [3] Complemented Groups. J.L.M.S. 12, 201-204, (1937). [4] On the Sylow System of a Soluble Group . P.L.M.S. 43, 316-323, (1937). 9. Huppert B : Endliche Gruppen 1., Springer-Verlag, Band 134 (1967). [1] Chap. I, 15.7, (97-98). [2] Chap. I, 15.8, 15.9, (98-99). [3] Chap. I I , 3.10, (165-166). [4] Chap. I I , 3.11, (166-168). [5] Chap. VI , §7, Formationen, (696-711). ' [6] Chap. VI , §11, §12, §13, (726-760). [7] Chap. VI, 13.7, (747-749). [8] Chap. I, 18.6, (131). 10. Maclain D.H. [1] The Existence of Subgroup of Given Order in Finite Groups. Proc. Camb. Phil. Soc. 53, 278-285 (1957). 11. Pardoe K. [1] An Embedding Theorem in Finite Soluble Groups. Masters Thesis, University of Warwick. (1970). 

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