Embedding Theorems i n F i n i t e S o l u b l e Groups by P e t e r W a l t e r Hughes B.Sc, U n i v e r s i t y o f Auckland, New Zealand, 1968 •M.Sc, U n i v e r s i t y o f Auckland, New Zealand, 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF S€*EWeE AfCTS In t h e Department of MATHEMATICS We a c c e p t t h i s t h e s i s as conforming t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA September, 1971 . & In p r e s e n t i n g an this thesis advanced degree at the Library I further for shall agree the in p a r t i a l fulfilment of University of make i t f r e e l y that permission s c h o l a r l y p u r p o s e s may by his of this written representatives. thesis for It financial available granted Uf[. If gain Columbia 7/. by the i s understood of The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a for for extensive permission. Department Date be British shall requirements Columbia, H e a d o f my be I agree r e f e r e n c e and copying of that not the that study. this thesis Department copying or for or publication allowed without my By a group we w i l l mean a finite soluble group. It is an interesting fact, (Pardoe [1]), that the subgroup closure of the class of groups Let P^, those with a unique complemented chief series, i s a l l groups. X be the class of groups with a complemented chief series. We investigate the action of closure operations P^. T such that TX = X upon The purpose of this i s to find a collection of such closure operations whose join applied to i s X . In the course of this investigation we introduce a new closure operation M defined by ; M/ = { G | G = <X --.,X >, X . V, 1} X ± n ± e sn G, ( |G : X j J . - . j G :X | ) = 1} n ACKNOWLEDGEMENTS The A u t h o r i s i n d e p t e d the s u g g e s t i o n t o h i s s u p e r v i s o r , Dr. T r e v o r Hawkes f o r o f t h e t o p i c o f t h i s t h e s i s , and f o r h i s v a l u a b l e a s s i s t a n c e and encouragement throughout i t s p r e p a r a t i o n . Dr. B. Chang f o r r e a d i n g t h e d r a f t copy. He would a l s o l i k e t o thank He i s g r a t e f u l f o r t h e f i n a n c i a l a s s i s t a n c e g i v e n by t h e Canadian Commonwealth S c h o l a r s h i p and F e l l o w s h i p Administration. F i n a l l y i t i s a p l e a s u r e t o acknowledge t h e p a t i e n c e , the c a r e and t h e p r o f i c i e n c y o f Mrs. Y.S. C h i a Choo i n t y p i n g t h i s thesis. Introduction 1 Chapter 1 §1.1 : D e f i n i t i o n s and N o t a t i o n 3 §1.2 : Some Known R e s u l t s 10 Chapter 2 §2.1 : Preliminaries 30 §2.2 : Two C l o s u r e O p e r a t i o n s 33 §2.3 : A Further Investigation of X ' 37 Conclusion 46 Bibliography 47 Introduction When given a group to know whether subgroups possess this property. manner. H G with a property of G, (*) i t i s often useful or quotient groups G/K of G, also We can pose this question i n a s l i g h t l y d i f f e r e n t 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 t h i s d i s s e r t a t i o n i s to pose a question of this type i n a formal manner and give some answers for c e r t a i n properties (*). More e x p l i c i t l y , given a class of groups T, i s there a simple description f o r the class X TX ? and a closure operation One way to tackle this problem i s by looking at T-closed classes of groups X and deciding whether or not, V that contain TX = Y. An obvious extension of this method i s to look at the j o i n 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. denote this class by We w i l l S. In Chapter 1 we w i l l discuss some known results of the type discussed. to 5 One measure of the complexity of a class of groups i s to look at i t s subgroup closure. If sX = S X relative then we can say X is " l a r g e " i n as contains several f a r as a l l groups can be embedded i n an X-group. Chapter 1 r e s u l t s of t h i s s p e c i a l type f o r w i d e l y d i s s i m i l a r classes X. In C h a p t e r 2 we discuss a unique complemented c h i e f We i n v e s t i g a t e whether the c h o i c e of c l o s u r e has not s e r i e s and s u i t a b l e s e t of c l o s u r e s i m i l a r classes of groups, t h o s e w i t h those w i t h a complemented c h i e f former can be o p e r a t i o n s that been answered, but two extended to the l a t t e r by l e a v e the some p r o g r e s s has o p e r a t i o n s t h a t may latter fixed. As yet a series. judicious this question been made towards s e l e c t i n g a work. §1.1. D e f i n i t i o n s and One Notation. o f t h e fundamental t o o l s used to generate new known groups i s t h e wreath p r o d u c t . groups from There a r e s e v e r a l d e f i n i t i o n s f o r the wreath p r o d u c t . The v e r s i o n we w i l l need the most i s the r e g u l a r wreath p r o d u c t and t h i s i s d e f i n e d as f o l l o w s . Definition : group The r e g u l a r wreath p r o d u c t , G wr. H, o f a group r G by a H, i s d e f i n e d by ; G wr. H = { ( f , h ) | h e H, f : G — > r H} where m u l t i p l i c a t i o n i s d e f i n e d by ; ( f , h ) ( f , h ) = (g,h h ) , 1 1 2 2 1 2 and g(h) = f ( h ) f ( h h ) 1 2 1 for a l l h in H. As we w i l l u s u a l l y be c o n s i d e r i n g t h e r e g u l a r wreath p r o d u c t we will drop the s u f f i x 1 r*. We w i l l need s e v e r a l r e s u l t s about wreath p r o d u c t s . (1.2.9) i s concerned w i t h e x t e n d i n g an automorphism automorphism of G wr. H. We of G, One used i n (or H), to an a l s o need two embedding p r o p e r t i e s t h a t a r e used e x t e n s i v e l y i n what f o l l o w s . (1.1.1) Lemma : Huppert [ 1 ] . (a). If a i s an automorphism o f G, so i s cT an automorphism o f G wr.'H where ; ( f , h ) " = (g,h), and g(k) = f ( k ) (b). If u i s an automorphism o f H, so i s u a for a l l k i n H. an automorphism o f G wr. H where ; (f,h) (1.1.2) Lemma : G^ wr. H^ y = ( g , h ) , and g(k) = g ( k u ) y Huppert [ 2 ] , for a l l _ 1 ( a ) . I f G-^ <_ G can be embedded i n G wr. H. and % k <_ H ( b ) . L e t N«4 G, i n H. then then G can be embedded i n N wr. G/N. Theorem a method (1.2.11) p r o v i d e s not only a r e s u l t o f i n t e r e s t b u t a l s o t o c o n s t r u c t groups t h a t have a unique complemented T h i s c o n s t r u c t i o n i s l a t e r used i n Chapter 2. chief series. I n t h e p r o o f we use two g e n e r a l r e s u l t s which we w i l l quote h e r e f o r completeness. The f i r s t result a l l o w s us to c a l c u l a t e t h e o r d e r o f c e r t a i n c h i e f f a c t o r s , w h i l e t h e second r e s u l t i s used t o p r o v i d e a c o n t r a d i c t i o n i n t h e p r o o f o f (1.2.11). (1.1.3) Theorem : Huppert [ 3 ] . L e t V module of dimension group A. Then A n over t h e f i e l d be a f a i t h f u l K = GF(p ), f irreducible f o r an A b e l i a n i s c y c l i c and t h e r e e x i s t s a group homomorphism ; 3 : A > GF(p )* , and a K-isomorphism, a : V > GF(p ^) such t h a t , a ( v a ) = 8(a)a(v) n f n + for a l l a e A and v e V. The i n t e g e r n |A| p n f (1.1.4) i s the u n i q u e l y determined s m a l l e s t i n t e g e r such t h a t - 1 . Theorem: Huppert [4]. Let G<_GL(n,p ), f A an A b e l i a n normal subgroup of G. L e t V = V ( n , p ^ ) , considered n module, be t h e d i r e c t sum o f Then G, considered as a group o f s = n/k as a p e r m u t a t i o n group on s e m i l i n e a r maps (u + u )fe = u-jg + u g , 2 c —> GF(p ) k f c& i s a field - l i n e a r maps o f (1.1.5) Corollary : Proof : Consider a t h e map k f a K[A] modules. a c t s i n the same way c e GF(p C^(A) I Also ) , s t n e s e t °f ). i s isomorphic a : g -> a -> Aut [ G F ( p ) k f = G a l o i s group Also as a K[A] e V(s,p k f V(s,p r i e V(s,p ), u V, ) , and such t h a t ; u isomorphism. G/C (A) G g, 2 ( u c ) g = ugcS, where i r r e d u c i b l e isomorphic GF(p^) 1 K = GF(p t o a subgroup o f where a g c —> : GF(p )] f [GF(p ) k f i s a homomorphism w i t h k e r n e l g i v e n by : GF(p )] f cg C k ' That i s , a = {g e G | a ker = {g e G | c = C (A) G Hence G/Cg(A) = a ( G ) , =1} = c 8 for a l l c in GF(p )} k f . a subgroup o f . A n o t h e r r e s u l t used i n the p r o o f o f (1.2.11) i s a theorem o f Dirichlet's of c o n c e r n i n g the d i s t r i b u t i o n of prime numbers i n a g i v e n sequence numbers. (1.1.6) Theorem : {m + kn | k = 1, 2, Dirichlet •••} [1]• If (m,n) = 1 then the s e r i e s c o n t a i n s an i n f i n i t e number o f prime numbers. We w i l l now i n t r o d u c e the concept o f group c l a s s e s and c l o s u r e operations. this These c o n c e p t s a r e e x t r e m e l y u s e f u l and w i l l be used throughout exposition. We say a s e t o f groups X d e f i n e d by some'group t h e o r e t i c p r o p e r t y p o s s e s s e d by a l l i t s members i s a c l a s s o f groups i f ; (a) . If G e X, (b) . The u n i t group 1 i s i n We w i l l then a l l groups i s o m o r p h i c w i t h A closure operation of groups s a t i s f y i n g ; are also i n X, X. denote by (1) the t r i v i a l c l a s s o f groups i s o m o r p h i c to G G, (with 1). c i s a map c l a s s o f groups and by (G) the from c l a s s e s o f groups to c l a s s e s (C.l).. X <_ CX, (C.2). I f X <_Y (C.3). CX = C(CX), We say X operations CX <_ BX. then expanding, monotonic, idempotent. i s C-closed i f CX = X. C <_ B The j o i n CX <_ CV, then If B and C are closure i f and only i f f o r a l l classes of groups {A,B} of two closure operation A and B X, i s defined by, {A,B}X = By AB(X) we mean n A(BX) {y | x < y, AY and we note that = B/ AB = y} . i s a closure operation i f and only i f BA <_ AB. An a l t e r n a t i v e method to describe a closure operation s p e c i f y the C-closed c l a s s e s . Let C C = {X ( CX = X} . Then the family (a) S e C (b) C C i s to be a closure operation and C satisfies; i s closed under taking a r b i t a r y i n t e r s e c t i o n s . 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 a r b i t a r y class of groups TX = X n {y T and i t s a c t i o n on an by ; | x <_ y e C} . Thus there i s a one to one correspondance between closure operations and f a m i l i e s 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 n {H G X} . E | K«d G e X} . QX = {G/K SX = <_ | H sn G e X} . N X = {G | G = <N , Q DX = {G | G RX = {G | Q D N >, X = H 3 X N r x ••. x H G, G/N ± , R £ X, ± H N sn X} . e X) . ± r H N i=l e X, ± ± = 1, i = 1, • • •, r>. Also the common classes of nilpotent and Abelian groups w i l l be denoted by W and Notation : If A, respectively. H G i s a group and H <_ G, H i s a subgroup of G, H i s a normal subgroup of G, H i s a minimal normal subgroup of G. H <3 oG, H i s a maximal normal subgroup of G. H sn G, H i s a subnormal subgroup of Core H , largest normal subgroup of NQ(H), the normaliser of C (H), the c e n t r a l i s e r of Z(G), the centre of G G. i s a subgroup of G we define ; G. H H in in G. G G. G. G. contained i n H . F(G). the F i t t i n g <)>(G), the F r a t t i n i Aut(G), the group of automorphisms <A>, the subgroup of = <H§ | g e G>, subgroup o f g ^ = n ^gh> G G. G. of g e n e r a t e d by a subset A of G. the subgroup g e n e r a t e d by the c o n j u g a t e s of in [g»h] G. subgroup o f H G. the commutator of g and h in G. [H,K] = < [h,k] | h e H, k e K >. G* = the commutator [ G , G ] , |G|, the o r d e r o f G. JG the i n d e x o f H : H|, IT(G), a H a l l Tr-subgroup o f i.e. , G. G , where l a r g e s t subgroup of the c y c l i c S(n), If a G. in G . the s e t o f prime d i v i s o r s o f G^, Cp subgroup o f group of o r d e r G and b a r e i n t e g e r s we a|b, a divides a|b, a does n o t d i v i d e (a,b), the g r e a t e s t i s a set of primes. whose o r d e r i s a p. the symmetric group o f degree a TT n. define, b. b. common d i v i s o r o f a and b. Tr-number. The f o l l o w i n g e x p l a i n s the n o t a t i o n used i n (1.1.3), (1.1.4). Zq, the i n t e g e r s modulo q, the f i e l d GF(p ), the G a l o i s GF(p the a d d i t i v e group of t h e n ) , GF(p )*, n f i e l d of order of q e l e m e n t s . p . n field. the m u l t i p l i c a t i v e group o f the f i e l d . G L ( n , p f ) , t h e group o f i n v e r t i b l e l i n e a r t r a n s f o r m a t i o n s o f a space o f dimension §1.2. n over vector GF(p^). Some Known R e s u l t s . In t h i s s e c t i o n we w i l l b r i n g t o g e t h e r some known r e s u l t s about c l o s u r e o p e r a t i o n s a c t i n g on g i v e n group c l a s s e s . It i s w e l l known t h a t the converse of Lagrange's theorem i s f a l s e . There a r e groups, f o r example, t h e a l t e r n a t i n g group o f degree 4 , which do not p o s s e s s subgroups o f a l l p o s s i b l e ' o r d e r s d i v i d i n g |G|. I f M denotes the c l a s s o f groups which have subgroups o f a l l p o s s i b l e o r d e r s it 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 for s o l u b i l i t y that M <_S. The f o l l o w i n g theorem shows t h a t , i n the sense t h a t any s o l u b l e group can be embedded i n a group i n M, t h a t the c l a s s S. (1.2.1) Theorem : M a c l a i n [1]. S = sM M i s l a r g e i n r e l a t i o n to Proof : Let G be a s o l u b l e group w i t h o r d e r r II i=l A b e l i a n group of o r d e r l 2 d = p^ p d d d P 2 such H that r i a.-l p. <^ d^ , i = 1, be a -Hall iT-subgroup of 2, -G and U of o r d e r subgroup Let U be M. is in an For, l e t . . [G x U|. • • •, s ; We can r e o r d e r the d^ < TT = {p^, p , '*", 2 g ^ B i s c a l l e d complemented i f t h e r e i s a a S r ••• p a n of J_ d s+l p . P }> • • •, r . ••• p a s" s , i = s + 1, d/s l ~ l p.. K of G H such G that, i n w h i c h e v e r y subgroup has groups . We namely t h a t prxmes p have G |G|. C <_S, d a d S T I G are HK = G and H f\ K = 1. a complement, i s c a l l e d i s s o l u b l e i f and o n l y i f i t has The class . Then | H X D | =d The c l a s s of groups the c l a s s o f complemented C i s both G Sylow p-complements f o r a l l S- and D ~ c l o s e d . 0 a r e elementary Also i f A b e l i a n and G e C the c h i e f factors cyclic. A subclass of squarefree order. That C i s the c l a s s of groups Q, the groups of these groups are s o l u b l e f o l l o w s from Burnside's Theorem, namely t h a t i f a Sylow p-subgroup i s c o n t a i n e d i n the c e n t r e of i t s n o r m a l i s e r then i t has a normal Sylow p-complement. H <_ G e Q has Let by P. H a l l ' s c h a r a c t e r i s a t i o n of s o l u b l e groups, then t h e Sylow p-subgroups of of G x U . 1 the p r o o f i s complete. A subgroup C then where d subgroup o f p. 1 be a d i v i s o r of r II i=l then |G| = a complement i n We now |H| m where ( | H | , m) = 1. Further, i f So, by H a l l G. t u r n to the c o n n e c t i o n between Q and C. [1], H (1.2.2) Proof Theorem : : [3]. SD (Q) = C, P. H a l l By c l o s u r e axiom Q Thus we need o n l y show t h e o p p o s i t e Let 1 ^ g e G G g. the d i r e c t p r o d u c t o f t h e groups be a normal subgroup o f Then G maximal Hg = G/Gg. in C G has t h e properties. (1) Every Sylow subgroup o f (2) Every c h i e f f a c t o r o f (3) G i s i s o m o r p h i c t o a subgroup o f We know by t h e above remarks t h a t a group The groups Q Q inclusion. and l e t with respect to not containing following SD (Q) <_ SD (C) = C. (C.2) and the above remarks H g have t h e p r o p e r t i e s G G i s elementary A b e l i a n . i s cyclic. (1) and (2) and a l s o the p r o p e r t y . o f c o n t a i n i n g a unique minimal normal subgroup. Thus i t i s s u f f i c i e n t t o show t h a t a group G s a t i s f y i n g (1), (2), and (3) i s i n Q. By (2) t h e unique minimal normal subgroup and has o r d e r p. Let C = C (N). G We show t h e r e e x i s t s an A b e l i a n normal subgroup and K/N (a). of K i s a chief f a c t o r of Suppose p ^ q. i s characteristic different from N, G. of G, By (2), K/N Then, i n K, K C = N. and K G c o n t r a d i c t i n g (3). N of G i s cyclic Suppose n o t . such t h a t Then N <_ K <_ C, i s c y c l i c o f o r d e r a prime q. b e i n g A b e l i a n , t h e Sylow q-subgroup has a m i n i m a l normal subgroup (b). Suppose Sylow p-subgroup o f p-subgroup o f of G G. G So c e r t a i n l y G As N isa w = K]F, So we can form n a t u r a l way. Then, as | K | = p^, K i s A b e l i a n by ( i ) , F = G/C (N) and thus Aut(N). p = q. N^J G and and as each i s c e n t r a l i s e d by each Sylow i s c e n t r a l i s e d by each Sylow p-subgroup p' group i s o m o r p h i c t o , a subgroup o f the s e m i d i r e c t p r o d u c t o f ( |K|, |F| ) = 1 use Maschke's theorem t o g i v e K < G, K = N x M, where and K K F , i n . the by i s A b e l i a n we can M<J G. This again contradicts (3). Thus N = C and G/N which i s c y c l i c o f o r d e r where 1. k p - 1. By (1), square o f a prime ; so i s i s o m o r p h i c t o a subgroup o f Aut(C ) , k G i.e. G i s metacyclic of order pk, c o n t a i n s no c y c l i c subgroups o f o r d e r t h e i s square f r e e and G e Q., completing the proof. N i l p o t e n t groups a r e c h a r a c t e r i s e d by b e i n g t h e d i r e c t p r o d u c t o f t h e i r Sylow subgroups. Thus i t i s always p o s s i b l e t o c o n s t r u c t a normal s e r i e s i n which the f a c t o r groups have o r d e r t h e l a r g e s t power o f each prime d i v i s o r d i v i d i n g t h e group o r d e r . the c l a s s o f Sylow Tower groups T = { G |G Thus n i l p o t e n t groups form a s u b c l a s s o f T, which i s d e f i n e d by, has a normal s e r i e s where t h e normal f a c t o r s 1 = GA G-, <d. •••<<! G Q G±/G±„j_ a r e n = G isomorphic to S y l subgroups o f G} . A normal s e r i e s f o r a group 14 0 (G)<3 0 p p I p (G) 0 G , (G)'<3 may be d e f i n e d by •••<G (1) ow where i f IT i s a set of primes subgroups of define 0^ G w G^CG) i s the product of a l l the normal whose orders are divisible only by members of r r-1' inductively by v (G), where TT . , 1 i = 1, We are sets of primes, • = r-1 The set of primes r TT. IT' 0_ /°Tr _ ...Tr (G) G r 1 1 A is just the complementary set of primes to Clearly for a nilpotent group the series stops at That is the number of factor groups in (1) divisible by p TT . 0pt (G) = G. p for each prime p is one. However this i s also true for Sylow'Tower groups as the series reaches G at most at °p'pp' ( )• Groups with this property of having G only one factor group i n the series (1) divisible by are called groups of L^'Cl) p length 1, and form the class p for a l l primes L (l). m can also be locally defined by the formation function The class S^ of a l l groups not divisible by T p p The class f : p —> S^,. i s {Q, S, R , N } Q Q 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 : Proof : Alperin [1]. SN T = It i s clear that C.2, SN T <_ SN Loo(l) = L o o ( l ) . Q 0 stronger result namely : T <_ 1^(1) L (l) m and so by closure operation axiom To show the opposite inclusion we prove the G e L (l) If L and We prove t h i s (a) . G e T, ± (2) and If G. TTCG/N-L) Tr(L ) N-^ and N TT(G/N ) = 2 (b) . G then 0 ?(G) G. normal i n G/N-^ * G / N = 1, P c o n t a i n s a subgroup i s o m o r p h i c 2 P and G e SD (SN (T)) 0 0 (2) . -n(.L ) <_ T T ( 6 / P ) . ± P wr. G / P and by Nilpotent In f a c t , N e N T common to them a l l . n S . 0 N. If G of is and Q So., <_ P wr. L = B L = B L ^ B L 2 (1.1.2), G e SN T • completing D B L . N E (2) N T, 0 p/ |L|. Thus the p r o o f . groups have the p r o p e r t y of b e i n g b o t h S n - and c l a s s e s of groups which a r e c l o s e d under t h e s e two o p e r a t i o n s a r e q u i t e numerous and have been s t u d i e d X =' { S , N , } X and Q t h e Sylow p-subgroup i s the base group of the wreath p r o d u c t and by Q = SN T, So by the i n d u c t i o n h y p o t h e s i s ; 2 N -closed. L |G| . has a unique m i n i m a l normal subgroup G / P = H <_ L - j L * • - L T T C B L ^ X £ TT(G) ± a r e two d i s t i n c t m i n i m a l normal subgroups of 2 (1) . B L <4 where TT(G). Suppose {p} = TT(N), = L <_ T T ( G ) . I So by the i n d u c t i v e h y p o t h e s i s U where G = H ^ L ^ - ' - I ^ a s s e r t i o n by i n d u c t i o n on then, as I s w e l l known, with (1) then, r a Thus we say a c l a s s X We w i l l c a l l a F i t t i n g c l a s s to d e t e r m i n e p r o p e r t i e s i s a F i t t i n g class i f F trivial i f F = (1) or For V of G, any class of groups and any group c a l l e d the X-injector ; i f f o r a l l N H V e X i.e. X Of course such a N sn G, V i s X-maximal. X to c a l l a F i t t i n g class F N. isa I t was proved by i s a F i t t i n g class, every group has a unique conjugacy class of X-injectors. G N (\ V may not e x i s t , but what x^e can say i s , i f F Gaschutz, Fischer, and Hartley [1], that when of we define a subgroup and i s not contained properly i n any X-subgroup of F i t t i n g c l a s s , then every group has an X-injector. Gp G I t now makes sense normal i f f o r any group G the F injector i s normal i n G. In a recent paper, Gaschutz and Blessenohl [1], investigate 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 F i t t i n g class. (1.2.4) C G-B ; Satz 3 3 . Consider a group Example : and l e t •••, M c y c l i c group of order r be the p p - 1. G with chief series chief factors i n C. Let A be a We define d-i(g) = determinant of the l i n e a r transformation induced on Let d (g) r r n d (g) i=l ± or d (g) = 1 G by conjugation with if G g. has no p-chief factors. Define for a l l Then the classes classes. F for each prime p are g in G } . non t r i v i a l normal F i t t i n g If a non t r i v i a l Fitting class a prime, then there exists some n in > 0. a l l integers X .'-.'.'/'racteristic of F contains a group such that mnG wr. C X For any class of groups G and is i n p p F, we define the by •char X = e X } { p I C . P F is a Fitting class this definition is equivalent to defining char F = {p | there is a il-'Us a normal Fitting class F ± 1, !:V.i- a l l primes p. G F in such that contains the class But the nilpotent groups W S p of p-groups are simply direct products N <_ F, for a l l normal Fitting classes p-groups, and hence p |G|} F ^ 1 . A non t r i v i a l normal Fitting class is a large class of groups, in the ;^!)se that i t s subgroup closure i s a l l soluble groups. \.|.; 2.5) Theorem : -*L»;gof : We may |G| . ^ G-. ^ 1 Let 1 ? F Let F < S. assume that oG G ^3 x and then there is a ^c-iHist monomorphisms y v G and |G : G j m Let = p. such that y 2 sF = S . be a normal Fitting class then G e S If and proceed by G-j^ = 1, e F . mGiwr.C then G e W <_ F . But then there such that y 1 > G-^ wr. G/G-j^ = Gj wr. C p induction 2 > m Gj^ wr. C p In g e n e r a l normal F i t t i n g shown i n the (1.2.6) following F^ Let (1.2.4) w i t h be the p = 3. to be where A = triples defined x C^ (a,b,y) has a chief Let the = B, M =1 form g if i s of ((u,v),x). with 3-chief S3 = A/B, 2 But So (a,b) be e A, We the inverting can y e X; class s e m i d i r e c t p r o d u c t of involution, an i.e. c o n s i d e r elements of and G multiplication is i ^2 Sg ((a,b) (u,v) ,z). y B « < ( ( l , b ) , l ) > = A<d then the M.^ the the ((u,v),l) form In a l l cases = G/B factor = ((a,l),l) > = B 4 A and the = M^. 3 are d (g) G chief 3-chief and series =2 d (g) is = d-^g), l«d = AX. Now if = 1 2 for dg(g) G- factors. d^(g) = d-i(g) X In t h i s c a s e g is of and G e A^^ S^ and F^. , d ((12))=2. 1 • T h i s example suggests the trivial Fitting F, i s the QF This i s series 141 < i Q-closed. by G d (g) G =2. ((a,b),'y)-((u,v).,z) Then not t r i v i a l normal F i t t i n g 4 w i t h an |x| and where non Let elementary A b e l i a n group of o r d e r G = AX are example. Example : described i n closses class ?" universal q u e s t i o n , "what i s I t would be class. QF for F a of i n t e r e s t to know whether, f o r non such F If = { H | H = G/Gp is f o r some G} . B l e s s e n o h l and Gaschlitz prove t h a t i f F a non t r i v i a l normal F i t t i n g c l a s s then R a d i c a l - q u o t i e n t ( F ) o t h e r words the F - i n j e c t o r s o f Moreover, (F) = i s a F i t t i n g C l a s s d e f i n e the c l a s s R a d i c a l - Q u o t i e n t G In always c o n t a i n the d e r i v e d group c h a r a c t e r i s e s normal F i t t i n g c l a s s . G' . on R a d i c a l - q u o t i e n t ( F ) they show t h a t 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 Radical-quotient(F) <_ A . Indeed, any s t r u c t u r a l r e s t r i c t i o n on F seems t o imply n o r m a l i t y o f the F i t t i n g c l a s s in the f o l l o w i n g p r e c i s e sense : (1.2.7) Theorem : S. d i f f e r e n t from is B-G. [1] Suppose a non t r i v i a l normal F i t t i n g (1.2.8) Theorem : F <_X. for F Then F be a S - c l o s e d c l a s s o f groups i s a F i t t i n g c l a s s and from 5. Suppose or F i s a non t r i v i a l normal F i t t i n g i s a non normal F i t t i n g c l a s s , then C = S F < _ X . . Then Let X be a Q - c l o s e d i s a F i t t i n g c l a s s and class. In other Sylow p-subgroup and any subgroup Sylow of o r d e r prime t o a i s c o n t a i n e d i n a Sylow p-complement. P . H a l l [4] has shown t h a t t h e s e p r o p e r t i e s can be c a r r i e d over t o Sylow systems. • ''T? r be a complete s e t of Sylow-complements ' i ' = S, C = Q. complements and these a r e c o n j u g a t e , p words, C(Radical-quotient(F)) In any s o l u b l e group the Sylow subgroups always have S]_, ••*,S F class. F.P. L o c k e t t . [ u n p u b l i s h e d ] . c l a s s o f groups d i f f e r e n t if F Let X e i n a group * One r e p r e s e n t a t i v e from each of t h e r G Let of order conjugacy c l a s s e s o f complements of Sylow subgroups. 2 intersections, r r A Sylow system c o n s i s t s o f a l l 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 subgroups. If normaliser, of S S i s a Sylow system o f a group N = N(S), into i t s e l f . t o be a l l those g A l t e r n a t i v e l y we The groups i n W we G that can d e f i n e Of some i n t e r e s t i s the c l a s s W = { G in G W can d e f i n e t h e system t r a n s f o r m each member r by, N = 0 Ng(S^). i=l N d e f i n e d by : | the system n o r m a l i s e r s of G are c a l l e d S-C groups. Huppert known p r o p e r t i e s of system n o r m a l i s e r s and S-C [6], groups. are s e l f normalising}. g i v e s many of the Carter [ 1 ] , proved the e x i s t e n c e o f s e l f n o r m a l i s i n g n i l p o t e n t subgroups i n any group and showed t h e s e were a l l c o n j u g a t e . F o r the results next r e s u l t we need two standard about the-system n o r m a l i s e r s o f a s o l u b l e group, namely ; (1) Every system n o r m a l i s e r c o v e r s each c e n t r a l c h i e f factor, (2) Every C a r t e r subgroup c o n t a i n s a system n o r m a l i s e r . and 3 • (1.2.9) Proof • Theorem : Huppert [ A l p e r i n Thompson] [7]. Let 1 = G <d G-L4 0 G e 3 * ' "4 S(W) and G N =' G = -S , • . be a c h i e f s e r i e s for G. Let Choose a prime p d i f f e r e n t k ± <_ n ± and (a) p" from a l l = l(p) . 1 = G^/G^_^ Let N ± the p / s = GF(p By a s i m p l e i n d u c t i o n , we show r e g u l a r wreath p r o d u c t ( • • • (N-^ non t r i v i a l has an automorphism p r o o t o f u n i t y i n GF(p Claim The group x U| 1 ^ h e H i n U^. i s fixed by.all c a Suppose t h a t c l a i m i s t r u e f o r By y of order (f,n) The i d e n t i t y , p = (f*,n ) y y ( for suitable (f,n) (f,n) y shows t h e automorphism so t h a t c a n be embedded i n t h e i t e r a t e d . As m wr. N ) " « ) wr. N ^ ] wr. N . 2 m of order p t h a t f i x e s no m u l t i p l i c a t i o n by a ) . with [u^.[ = p , _ 1 y = and such t h a t no k For k = 1 s e t U-^ = <a > . and c o n s i d e r the group (1.1.1) t h e r e corresponds t o t h e automorphism automorphism 1 "H^. = (•••(Nj wr. N2)''*) wr. N^. has an e l e m e n t a r y • A b e l i a n group o f automorphisms K n^ |K^| = p ^ . . elements ; namely t h a t i n d u c e d by l e f t primitive : G + > [(•••(% m Every ) n ± and and choose wr. N2)**') wr. N G - — > G _2_ wr. G / G J J J . ! (b) i = 1, • • *m of + = B^. wr. of a n H ^ + i d e f i n e d by where g y = ot^ j H]£+;L f'(n) = and 1 f f(n y _ 1 ). n e N^ ^), + (g.n^n ) -1 a c t i n g on (1.1.1) we c a n d e f i n e t h e automorphism H^^/H^ w r . l , f i x e s no element. cT e A u t ( H ^ ^ ) + By corresponding to N^-j.- a e U is by k (f,n) = a (f',n), where a group o f automorphisms of H -^ pk l. and T = a E Ujj , (c) chosen Let p we N (U ). H x e C (U H for a l l T ) n H m So H ^ in U -[_, ) = 1 m H ^ H, Hence thus covers normaliser i s contained normaliser for By H (a) U, m and for a l l a i n N (U ) = U H m as permutation r e p r e s e n t a t i o n . the F i t t i n g subgroup. is those c+ p C> and k by order T = y" by U m . We have [8]. Let [x,a] e PL^ f\ U U m being Abelian i s H covers m of = 1. m a l l central i s A b e l i a n . A l s o a system m Thus to a subgroup of U m i s a system H a S-C-group. H, completing the proof. a faithful primitive T h i s i s e q u i v a l e n t to s a y i n g the group has I t t u r n s out t h a t t h i s a 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 that possess a s e l f the c l a s s o f p r i m i t i v e groups <_ k U, and m H/IL^ = U complemented unique minimal normal subgroup. U> L e t Uj -^=<U| [I> . so can a p p l y Huppert i n each C a r t e r subgroup. i s isomorphic is | a e considering of which i s s e l f n o r m a l i s i n g making G \={a Thus the c l a i m i s p r o v e d . A group i s c a l l e d p r i m i t i v e i f i t has PQ = { G | Q(G) . then by k+ thus a C a r t e r subgroup of H. A system n o r m a l i s e r c h i e f f a c t o r s and Then i s e l e m e n t a r y A b e l i a n of k + f = n = 1. ( |Hj,|u | = 1. . to the semi d i r e c t product Then as m T see t h a t H = HjjjlLjj such t h a t x e HJJ, H So (f,n) a i s elementary A b e l i a n o f o r d e r a e U^. commute. Also i f + isomorphic k+ By our i n d u c t i o n h y p o t h e s i s d e f i n i t i o n y and f*(x) = f ( x ) c e n t r a l i s i n g m i n i m a l normal subgroup. P. A s u b c l a s s of P, i s defined We call by P } . Alternatively can be d e f i n e d as the c l a s s of groups w i t h 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 a Suppose <j)(P) = 1. P Hence i s a p-group i n pQ. F(P)/<|>(P) = P, Then as P i s i n P, we have i s an elementary A b e l i a n p-group. But the only e l e m e n t a r y A b e l i a n p-groups i n P^ a r e those o f o r d e r q, q a prime. Let complemented. (1.2.10) X be t h e c l a s s o f groups i n which every c h i e f f a c t o r i s Then we have the f o l l o w i n g Lemma. Lemma : If G e X G, and then N i s complemented i n G. Proof : N <_ H. We use i n d u c t i o n on I n (2.1.1) we show H/N<^jG/N, L/N chief factor N such t h a t H (M n M fi L N(M H M in G. L) = 1 L ) = HN(M H L ) = HL = G ; i s a complement Clearly P^ < X, H/N of H so i f L/N H We c o n s i d e r (M n such that and | G / N | < | G | . As and f l L = L, and N ° ^G, and i s complemented L/N • H/N = G/N has a complement Dedekind's modular law, So so G/N e X QX = X, by the i n d u c t i o n hypothesis, there e x i s t s Also | G | . L e t H-4 G and L) fl i n G/N. H/N = N/N. M O N n L. The By (M (\ L) = 1. H = M H N = 1. in G. E<§ G e pQ, then N i s complemented i n G. (1.2.11) It i s of interest to know what t h e subgroup c l o s u r e o f Theorem [1]. Pardoe i.e. S(pQ) =5 pQ i s . Proof : We w i l l |G|. i n d u c t i o n on monomorphism, in pQ g i v e a survey Let prime p ^ q or r. W = W/N = C p p wr. C q Let in S H'-<3 M = M, NJ; = M| , Let S (1) L (2) T E PQ, T = SC . p, and We w i l l p We choose a and l e t W = M wr. C N = NjX.-.xNp, can be embedded i n refer S provided ± _ c a l l the c o n s t r u c t i o n of t o i t as such later on i n t h i s S and, G | h Now T = SCp i s clearly must pass through ± e H } £M-[X...xM t The p r o o f of p N. , a S/CgCH^) = M / C a : 'sCgCH^ = ( s M l s (Hj.) for a l l • • •, s , • • •, s ) C ± p i = 1, - . - . p . g in P^. For l e t , Then, i contained p j q - 1. = {(l,...,h ,.--,l) . W under t h e n a t u r a l p r o o f o f (1) i s r o u t i n e and w i l l be o m i t t e d . i then I t i s then shown t h a t , We c l a i m t h a t any c h i e f s e r i e s o f where = C^ . T i = l,---,p, (2) c o n t a i n s t h e c r u x o f t h e argument. M M/M be t h e i n v e r s e image o f W. D e f i n e p r o o f and i n Chapter '2. where As t h e c h i e f f a c t o r s o f groups i = 1, 2, T, "Pardoes C o n s t r u c t i o n " , and w i l l \ By i n d u c t i o n t h e r e i s a be a minimal normal e c c e n t r i c subgroup o f : W —> The and 0 We w i l l use . t h e base group. homomorphism M e P^. G E S. Let G / L = Gy. and where Let If p M, centralising, = (M x. • . x M ) C . x L<3">G u : L —> are s e l f o f t h e method. (IL) —> s ^ O L ) . p is the r e q u i r e d The f o l l o w i n g isomorphism. diagram Let o f normal K«£2,T subgroups such t h a t , i s then K N and N ^ K. applicable. KN K N < KN<3 T, As N then K N >_ S . N/(KHN). centralises K Thus centralises between K H N and N. Abelian C (N/H) >_ N and combining Let N/F = T H^-<1 N.<lM. ^ x • • • x Let N / H N factor t h e s e two r e s u l t s , of the chief K / ( KH N) so a l l chief be a c h i e f be t h e top p o r t i o n factors of T. of ^NK>_S. T series N/H is As C (N|H) T f o r M^. Let i N^ x ... x H. chosen as [ K , N ] <_ K f) N , Also x N x... x R_, From what f o l l o w s P H so below, H can be ± N/F e Q(N/H) and hence C, (N/F) >_ C (N/H) _> S. r T N. Then 1 = S/Cg M /C _ (N /H ) = — i H i Cj^_(N^/H.) = N.. i i x if K <J T M L i i i 1 . For, M e P\ ± giving . Thus no such then e i t h e r K <_ N K can e x i s t . or T h i s c o n c l u d e s the p r o o f that N <_ K. Next we show i f , 1 = M. , < M. . _<3 i,k i , k - l is the chief is a chief assume t h a t sufficient series f o r M^, f a c t o r of the chief t o show A T, then when series A^ x M. „ <3 N. <! M. i,2 i i /M M_. . x ... x M ./M. ... x i,J p , j i,3+1 j >_ 2. for F o r , by i n d u c t i o n on i s , 1<1 A<3 N^<3 M^, x A p o<J T. x •• M ^ p »j+l | G | we can and i t i s T h i s i s done by n o t i n g that, U»4 if T and U <_ A, then A^ <_ U The e s s e n t i a l p o i n t where we have assumed Z f then we q N S-submodules where Q Then S q i N.«<1 M. PQ groups i n . S/N = S If S. |N^|, £| then the i s an elementary A b e l i a n i s a Sylow q-subgroup of is a faithful A <_ U. N £ Choose C„ Let i n t e g e r n, such t h a t plq - 1. If S = NQ pj q - 1. such t h a t and by T, a prime, dim (1.1.3) (S)= 7 P = smallest l are i r r e d u c i b l e q-group. a p i r r e d u c i b l e module f o r 1 1 o < : a r e c y c l i c of prime o r d e r . as a S-module under c o n j u g a t i o n , and and hence i n t h e argument comes i n showing t h a t i n d u c t i v e l y that as the o n l y consider f o r some \ dim„ (S) > 1 , Hence — ~ and as 2 S = Q i s an elementary A b e l i a n cyclic. Consider N q-group o f o r d e r as a Q-module. at least By Maschke's q , theorem Q N i s not i s a semisimple Q-module and, N is the d e c o m p o s i t i o n of Then N N = N-L © N 2 Q/Cq(N ) is cyclic. i Thus H ± f Z 1. = Nj to permute t h e C p = <g> CQ(N ) 1 So the = . But f o r some N^, g "S = QN/N O f so, Q N C Q ^ X J U by the remark f ± Then is a and module and by 1 Q = M^/C^OL) so t h e r e (CQ(N£))§ N. i ^ j . T, = in s Let (Q/CQ(N )) Also S/CgCH^ Suppose , p i n t o i r r e d u c i b l e Q-submodules. i s a faithful irreducible ± • • • $ N . C , = Q /N g H (1.1.3), that . • The a c t i o n of such t h a t p CQ(N^). = Q§ = Q/N = Nj H c o n t r a d i c t i n g the f a c t a r e non i s o m o r p h i c i n p a i r s and hence t h e o n l y C is and Q = Q . that p Hence "s«< submodules f o r T . N a r e d i r e c t sums o f the contains So a N^ N^s , i =].,•••,p.. and by the a c t i o n o f permuting the p N^, submodule contains N. N«<]T. To show T 1 <^J • • • <3 series is in M-^ 2 i t i s s u f f i c i e n t to show t h a t t h e c h i e f x ^ x p chief factor i s self centralising. The f i n a l s t e p (1) Suppose r f 2 . f? and T i n s t e a d of of T and p replaces primes p such t h a t q. G Now Suppose ^ S < ^-'- ^ s complemented cases. the above c o n s t r u c t i o n w i t h r i n s t e a d i s the maximal normal subgroup By (1.1.6) t h e r e are an i n f i n i t e number o f and r j p - 1. The new group U o b t a i n e d > L wr. G/L > S wr. C r = 2. Since > U <_ T wr. r t o M wr. C^. that s j q - 1 suppose and Let 5|q - 1. 5 j s - 1. X C r U e P^. 2|p - 1 f o r a l l odd primes p, a t t h i s Without l o s s of g e n e r a l i t y we can assume ' 5 { q - For, contains (1.1.2) s t a g e we w i l l have to r e s o r t to a d i f f e r e n t method o f p r o v i n g construction and each These a r e r o u t i n e and w i l l be o m i t t e d . T' = S c a n be- embedded i n the group (2) N < p = 2 + n r , (some n ) , so i t i s p o s s i b l e t o choose a as a subgroup and by G < We repeat M. prime p such t h a t p j q - 1 So 2 ^ i s b r o k e n down i n t o two of S C Thus any non t r i v i a l 1 be the group i n Then by and apply P^ (1.1.6) t h e r e Then we would a p p l y that "N°<lT". Pardoe's so formed. i s a prime s such Pardoe's construction to M wr. Cg to form a group construction to proof with A wr. C5 A e P^ . t o form Then we would a p p l y Pardoe's B e P^ . We would t h e n complete t h e B i n s t e a d o f X. A p p l y Pardoe's c o n s t r u c t i o n t o X wr. and -3115"^—1 . c o n s t r u c t i o n to If Y i s t h e group i n Y wr. C P^ so formed apply Pardoe's to form t h e group 2 31J5-1 and n o t e t h a t Z. These c o n s t r u c t i o n s give us a s e r i e s of groups w i t h normal s e r i e s . (a) Y (b) T C c 31 c Z (c) U V 31 3 c UxU=R 6 S II 1 R e c a l l t h a t f o r t h e c h i e f s e r i e s below R = R/S and "z = Z/S. We c a n c o n s i d e r and a and Y, D-module. t o show Note R 6 over c o n s i d e r e d as a 5"^. So R] = X # Y Suppose The o n l y p o s s i b l e i r r e d u c i b l e submodules A i s A b e l i a n and D e P , is i s o m o r p h i c t o a subgroup o f i r r e d u c i b l e module f o r Q D. then Thus C ( A ) = A. D Cg. R Z^ A = V/R , module i s t h e d i r e c t sum of two i s o m o r p h i c i r r e d u c i b l e of order "N°<3 T". R»< Z . as a v e c t o r space o f dimension D = Z/R-module by c o n j u g a t i o n . ( = C^), X Then i t i s s u f f i c i e n t R we j u s t use t h e method p | q - 1 was o n l y used t o show of t h e theorem, as t h e f a c t t h a t Let S modules i s n o t an i r r e d u c i b l e a r e X o r Y. Thus by (1.1.5) As C T h i s i s i m p o s s i b l e and thus 2 A<J D, = D/C (A) D R i s an R<»<3 Z . T h i s completes t h e p r o o f t h a t Z and hence Z belongs to P^. To complete (2) we note t h a t t h e r e a r e monomorphisms G which completes >- L wr. G/L > S wr. C the p r o o f o f t h e theorem. r > Z §2.1 : Preliminaries. In t h i s and s e c t i o n 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 the l a r g e r c l a s s of groups X defined by X = {GJ every c h i e f f a c t o r o f Carter, Fischer, and Hawkes [1] G i s complemented} have shown t h a t i f a group G . has a c h i e f s e r i e s a l l o f whose f a c t o r s a r e complemented then every c h i e f f a c t o r i s complemented. Thus we may X = {G| G define has a c h i e f s e r i e s i n which the c h i e f are complemented} If y . i s the c l a s s of groups w i t h t r i v i a l F r a t t i n i subgroup t h e n may be c h a r a c t e r i s e d X = { G (1.2.11) that every s o l u b l e X also by | Q(G) £ V } =" { G j F o r a l l Theorem factors K<J G, <J»(G/K) = 1 } shows t h a t the c l a s s group can be embedded i n a P^ P^ i s l a r g e i n t h e sense group. What can be s a i d about t h e groups t h a t can be subnormally embedded i n a P Q group ? pQ <_ X we have S P°- <_ S X n n = X. However we s h a l l see t h a t As S P°- < X . n are l e d to ask whether t h e r e i s a n a t u r a l c o l l e c t i o n o f c l o s u r e We operations whose j o i n a p p l i e d closed to e n l a r g e s i t t o X. Clearly X itself under each o f t h e o p e r a t i o n s a p p e a r i n g i n t h e c o l l e c t i o n . must be We first examine t h e s e . Proof X = {Q, Lemma : (2.1.1) : (ii) L e t H<3 G e X. = H < G in (L series L fV H of H)N* of G Suppose N , N <3 G , assume For, l e t x G. .G/N^N-j/N = Gl^ 1 e RX Q As Now G/N e X ± E X as a H-module. If L H-j/H ^ By C l i f f o r d ' s Theorem If N / H ^ o ^ i n H. i n H. complements N* < H. — Thus Hence H So t h a t has a c h i e f H e X i/H^_^ H/H.^ , and series S X = X. % H G and N 2 = 1, such t h a t N N/N 2 and G / % , G / N N <_ Nj_ : 2 n £ X . which have t r i v i a l 2 ? <f> (G/N- ) = ± 1, has a c h i e f 2 We may Now c o n s i d e r G/N. intersection. and G/N/N N/N = G / N _ / N N / N £ QX = X. X H Q 0< then G/N-^ as a member o f QX = X. G H. |G/N[ < | G | i t I s a l s o and as = H<! running through N/IL ^ N d N-^/N clearly i s a group o f m i n i m a l o r d e r i n R X - X . 2 has normal subgroups G/N r f a c t o r i s complemented. Then t h e r e e x i s t N-L«><} < H A i s a complement (iii) Also G X H-/H. , = N/H. .. x N /H. .. w i t h i l - l. l - l i - l i n which e v e r y c h i e f It < x d e f i n i t i o n of Suppose complements N < H., then — i ' H H i s completely reducible then with Q . Q Q From t h e a l t e r n a t i v e of a chief H^|lL_^ R , D }X n (i) 1 be p a r t S 2 Thus i n X. i = series 1,2; and <|>(G) <_ N x f\ N i n which e v e r y c h i e f 2 = factor 1. i n which e v e r y c h i e f f a c t o r i s complemented. (iv) If are r e s p e c t i v e complemented H and K = H < 1 = K <1 Q G e X and RX H]_ 4 Q • • • -4 H r =H ••• < K s =K chief series f o r H and K i n which each c h i e f f a c t o r i s then, 1 = H <f E <i Q ± .is a c h i e f s e r i e s f o r • • • <1 H < J H r H x K x K]_ <3 r •• • < q H r x K = H x K s i n which each c h i e f f a c t o r i s complemented. The case f o r t h e d i r e c t p r o d u c t o f a f i n i t e number o f groups i n by a s i m p l e i n d u c t i v e The c l a s s degree 4 i s i n X s t e p and hence X DX i s not S - or N D D - closed. The symmetric group o f i s t h e normal p r o d u c t of two K l e i n 4-groups and so and Cg. N 2 = Cy"C^ §2.3 we show of a c l o s u r e closed. Then follows b u t i t c o n t a i n s a subgroup i s o m o r p h i c to t h e d i h e d r a l Consider the semidirect group X = X . Q group D o f o r d e r 8 ; a group which has a non t r i v i a l F r a t t i n i Also = X. Q X and are i n 1 Thus G product G of Cy G e X . This group D e N X 0 X . N-j_ = Cy«»C2 ' and which have coprime i n d e x i n G _ by i t s automorphism i s the p r o d u c t o f two normal subgroups, which a r e i n subgroup. G. In o f f e r s on i n s i g h t i n t o t h e i n v e s t i g a t i o n o p e r a t i o n l e s s g e n e r a l then N Q and under which X may be §2.2 Two Closure Operations . The example of the holomorph of operation T C^ suggests we might define an on group classes by TV = (G | G = N N ; 1 2 ( |G : N |, x N ,N < G; N N 1 2 l5 |G : N | ) = 1} 2 2 e X ; • This definition, however, does not give us a closure operation as i t i s not idempotent. in §1.1. collection But recall the alternative method of defining a closure operation It i s sufficient to specify a l l the T-closed classes and that the C of T-closed classes satisfy ; (1) S e C ; (2) C i s •closed under taking arbitary intersections. We w i l l 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 w i l l denote by T. This description is in fact identical to one given by Kappe [1] when we restrict our attention to the universal class The next question to ask i s what i s the action of class of groups V. T S. on an arbitary This question has not yet been answered but i t s 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. this we define M by; To overcome nY = { G I G = < X |G : ( This operation G in G e M(MY) y. to Also i f (x p| 1 1 ,''',x ,X. > x j= | Y ± ) = 1. n r |G : X jj one ± x.. o f a s o l u b l e group L 5 a r e subnormal i n Y . : X ^ |. then and i f and M such t h a t specified. This one o f t h e P^ of M = G » S we may say G i i that and j . Then there i s a t i . This ^ j * * * , be a r e p r e s e n t a t i v e P^ <_ H^ . PJ_||G|, contradiction H r a r e subgroups i = l, then s e t o f sylow , , - ,s. Then p^ there Thus t h e remark i s e q u i v a l e n t t o where t h e o r d e r i n g f o l l o w s by i n d u c t i o n on contains But the X ^ r f o r each d i s t i n c t prime P i - ^ " * *^s and b e l o n g ( | G : H^|, •••». |G : H | ) = 1 G such t h a t n we c l a i m Bor a given p|y. f o r a l l (y^,•••,y )=l. i s idempotent. , i a j ) = 1. i r f o r a l lrelevent [2] has remarked t h a t i f H-^, G then x ^ = |G : X^ | , p, so subgroups o f that ( x ^ , ••• ,x p | x.. L e t P-^, P2,** ,P saying N where t h e X.. . r I t remains t o show i t i s • • • Y > where t h e Y . ^ a r e subnormal G = H^H2'**H f o r each x j ) =• 1 '.} y^ = | G : Y ^ | n o t d i v i s i b l e by Philip Hall e V ; ± A l s o i f we l e t F o r suppose p r o v e s our c o n t e n t i o n is |G : = | G : Y ^ | | Y ^ : X^j | = y ^ x ^ j * i least ,,i G = <Y and subnormal i n G are i n X X-J,- ± operation. then and b e l o n g t o M^. Y . = <X.,, Each n i s c l e a r l y expanding and monotonic. idempotent and hence a c l o s u r e If • • • ,X > ; X s n G ; X p of the product i s not | G | as a p r o p e r normal subgroup. G being s o l u b l e means Thus i n the d e f i n i t i o n i s t h e p r o d u c t o f subnormal subgroups and n o t merely g e n e r a t e d by subnormal subgroups. The c l o s u r e o p e r a t i o n group o f o r d e r 8 i s i n NA M TT =.{G | G = X X --'X 1 (2.2.1) Example. construction 2 , ; X n sn G ; X ± e V ; ± ^ n M TT = { a l l primes} and when i s c l e a r from the d e f i n i t i o n inequality i s s t r i c t . as the d i h e d r a l Q |G : X-L j , • - -, | G : X | ) i s a Tr-number} i s the empty s e t we get It N by d e f i n i n g ; ( when l e s s than to a f a m i l y o f c l o s u r e o p e r a t i o n s IT i s a s e t of primes, MY i s strictly but n o t i n MA . Q We can g e n e r a l i s e where M that T <_ M, we g e t and i n f a c t N . Q this For consider ; Let A (1.2.11) from and C B be groups i n P^ wr. Cg and 2 C7 wr. C5. formed by Pardoe's That i s A has a • 2 unique m i n i m a l normal subgroup of o r d e r 2 w i t h f a c t o r group c y c l i c o f o r d e r 3 and B has a unique minimal normal subgroup o f o r d e r 7^ w i t h f a c t o r group c y c l i c o f o r d e r 5. D be a subgroup o f subgroups o f the o t h e r hand B Let be a subgroup of o f o r d e r 7. G = A x B , and s e t V i s T-closed. p r o d u c t o f two members o f 1 n e i t h e r of these i s K = NjjN = A x B = G. 2 i s excluded. C V Let But N-^ = A x D V ='•{!, N-j_, N ) . 2 F o r i f K e TV then o f coprime i n d e x i n as i n t h i s case and N 2 A K. o f o r d e r 2 and l e t and Then K Hence T < M . 2 = C x B K = Nj_, i = 1,2, be G e ViV - / . On i s the normal We may suppose and a r e n o t normal i n G But t h i s has exhausted a l l the p o s s i b i l i t i e s c l a i m stands t r u e . N K e y that . Then so t h i s case f o r TV and our (2.2.2) Lemma : generating Proof is then we may choose t h e G e '^n where t h e X^ | G : X^ | = = o(x j_)' , : G. and n Let G = subgroup o f of Y = SV normal i n G. : and i f TN If then (x-^, • • •, x ) = 1. s e t of primes. i . e . H. = G . L e t H^ TT^, Then by t h e d e f i n i t i o n o f subnormal i n G, L e t cr(x^) = {p| p| x^} and 11 the complementary x^. a r e i n Y, H^ I t i s a w e l l known f a c t be a H a l l T T ^ subgroup i s a Hall that i f a H a l l Ti-subgroup c o n t a i n e d i n a subnormal subgroup t h e n so i s t h e j o i n o f a l l i t s c o n j u g a t e s . Hence <G^> £X. . Thus > <_ Core X Thus i f x^ = |G : Core X^|, = Core X]_. Core X - we have ••• Core X 2 prime d i v i s o r s o f . n |G : G'| . ± and a ( | G : Core X | ) = a (xj_). i (x-|_, • • • , x ) = 1 . n L e t p , i = 1,2, r, i Then t h e r e i s an x^ We now show G = be t h e d i s t i n c t such t h a t p. | . Q If G„ i i s a Sylow P^-subgroup 2 such t h a t G' <_M. r < G = G. P <_ M. But f_ ± - > x 7 r < Then I f n o t , then If p> <Gq> = P^ <_ P <_ M P = G. V •= S / , Core X^ t h e subgroups (x-^, • • • ,~x ) = 1. n Corollary P^<3 G P < G (G : M) = q, c o n t r a d i c t i o n and hence (2.2.3) then P. = <G i G P = P P "-P n G, > <_ x.^ . by t h e same 1 reasoning that 1 of then and q = pj ; some If i V, normal i n G, V = { S , Q}V then 0}lY = since This i s a G = <Core X | i = l , - - - , r > are i n n M<3 «G j <_ r , q j | G : M|. T h i s complete, t h e p r o o f . : We c l a i m and hence t h e r e i s an and t h e r e f o r e Thus P^ <_ Core X^. and as and s a t i s f y Proof : If V = SY we N ± = |G/K : X K/K[, A Further The Lemma Proof : We : MX = n Let e X < ± : 2 G/K |G : G e Ml'. X ^ / K ^ G / K X | ± G, , and so i f x = ± KY. G/K e Thus As are normal i n have ± X. the m o t i v a t i o n o f § 2 . 1 we the end we X K/K\ n was X^ where the In |G/K and f o r i n v e s t i g a t i n g the s t a t e d i t was in X. We now G , be and has .. X and subgroup of an = X n so by X < f G, e X, (2.2.3) we may and if x a group of minimum o r d e r in MX -X. QX ± = X ± by X^ thus ( |N|, Then G SX ± = assume ; i f |G. : X | G e MX, then I ± in X G group i s complemented i n X^ contains |G : X j ) = 1 contained X^ . But N = P , If 01 N and (1.2.10) t h e r e f o r e by G and (2.2.3) there a l l the p~subgroups of i s complemented i n and N«<3 Let the i n d u c t i o n h y p o t h e s i s and a complemented c h i e f s e r i e s . pjx^ subgroup of X 1. |G| and such t h a t N <_X . l 5 At = x G/N , N (x x ,•••,x ) = 1. have shown (x]_, • • - ,x ) N M. 2 |G/N| Also i n v e s t i g a t i o n of G = X]X "' n as V. K <3 G where claim. (2.3..1) then, = holomorph of C-j closure operation prove t h a t 2 then ± §2.3. G = X-^ X ' " ' X ( x ^ - - - ^ ) = 1. OY as H = G/K then assume and V in X K/K can Y, b e l o n g to QKY, He i s an A b e l i a n then i s an G. Hence normal shows t h a t every normal G. X^ e X implies Gaschutz's theorem [ 1 ] , that N x^ is complemented i n G. So G has a complemented c h i e f s e r i e s and thus MX = X. We w i l l now i n v e s t i g a t e t h e a c t i o n on of those c l o s u r e o p e r a t i o n s t h a t l e a v e X PQ fixed. o f t h e j o i n s o f some But f i r s t we w i l l i n t r o d u c e a u s e f u l normal s e r i e s . The Upper F i t t i n g 1 is D the length V(G/F _ (G)). ± of t h i s s e r i e s the F i t t i n g t h e upper F i t t i n g Lemma : I f H s n G e PQ F Proof : i ( H ) / F i - l ± ± length of G. s e r i e s c o i n c i d e s w i t h t h e unique such t h a t ) E S ( F i ( G ) / F i - l ( G we have X - 1. ) ' i = 1,-",A(G>. £(G). I f L e t l(G) = I. A l s o i f H sn G induction hypothesis ) l(G) = 1 t h e So suppose t h e h y p o t h e s i s i s t r u e f o r a l l groups F ( H ) = F ( G ) fV H i = H £(K) = I - 1. G/F-j^G) e QX = X. X ( then The proof f o l l o w s by i n d u c t i o n on result i s clearly true. for F (G) / F ^ C G ) = series. (2.3.2) as = G £ and D £(G) = £ a group i n P^ chief K 1 F (G)= 1 G, . . . <3 F ( G ) F ( G ) < ! F (G)<1 d e f i n e d i n d u c t i v e l y by We c a l l For = S e r i e s of a group then Then £ ( G / F ( G ) ) =1-1 11F (G)/F (G) 1 1 H F ( G ) / F ( G ) = H/F-jiOl). 1 and 1 1 sn G/F (G). 1 But Thus by t h e F ( H / F ( H ) ) / F _ ( H / F ( H ) ) • e S [ F ( G / F ( G ) ) / F _ (G/F (G) ) ] j L 1 i 1 But f o r any group 1 K , ± 1 i 1 ][ F (K/F^K))/F _ (K/F (K)) = F i i 1 1 2,''-,Z and the lemma i s s a t i s f i e d f o r i = by the remark t h a t (2.3.3) in Corollary pQ then Proof : H If : but Example Q n trivial H A b e l i a n then Sn(P^) < X . X isR G sn H normal subgroups i H-^ H = GH /H i ± Q P H Example R ^, 0 and R - c l o s u r e . Q F ( H ) . = H. : G G. Cp. x C3 The A b e l i a n group is in then of ± For, i f V H has P^ . G. e V . R S (pQ) < n isin r normal subgroups is in V . H.^ with Consider the r They have t r i v i a l i n t e r s e c t i o n and Thus X. V. i s a c l a s s o f groups and H/H^ G Q K X n condition f o r R S Q (2.3.2) By 1 - and S ~ c l o s e d , and, i n f a c t , we can show where sn H/H This i s a s u f f i c i e n t (2.3.5) i s covered i s elementary A b e l i a n f o r some prime p as i t i n t e r s e c t i o n such t h a t each G/'(H nG) i = 1 i t cannot be a subnormal subgroup o f a group i n then o The case i s an A b e l i a n subnormal subgroup o f a group and i s a closure operation. n G e S R y, in P F-^(G) : We know Q ± i s an elementary A b e l i a n p-group f o r some prime p. and (2.3.3) by (K)/F (K) F^(G). the unique m i n i m a l normal subgroup o f (2.3.4) R S I f -H H sn G e F^(H) <_ F-^(G) is F^B.) = H H . i + 1 isin R S ^ 0 n and S R n Q <_ R S . Q t o be a c l o s u r e o p e r a t i o n . n First observe t h a t , i f K i s a group T h i s f o l l o w s from the d e f i n i t i o n o f C o n s i d e r t h e bolomorph n G of Cy. Then G is in P P (\ X and the upper F i t t i n g S (pQ) n by series for G is l < Q C y < 3 G. (2.3.2) as the f a c t o r s i n i t s upper F i t t i n g elementary A b e l i a n p-groups f o r v a r i o u s primes p. by the i n i t i a l observation, not in Q n a p p l i e d to P^. But G i s not in s e r i e s are not a l l Hence G is in X but, R S (pQ). D n Another l i n e of a t t a c k c o u l d be D S Thus to c o n s i d e r the c l o s u r e operation 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 : Cg by x transform not 2 (a,b) S 0 n^^^ • L e t G b e t h e s e m i d i r e c product t x of in G is in Cg i t would be n * < where the a c t i o n of the g e n e r a t o r x to the d i r e c t p r o d u c t of any D S (P^), 0 C , D in (a~"'~,b -'-). Then _ two p r o p e r subgroups. S (pQ), which by n So, C if (2.3.2) i s not i s to 2 X G and pQ. To d e c i d e whether c e r t a i n subclasses of X are i n C l e a r l y the s i m p l e A b e l i a n not But elementary A b e l i a n , then I n what f o l l o w s we notation. = G 1 x G 2 If x G x QS (pQ). possible. n , where X, can b e g i n by such s u b c l a s s and as DX For i f <j>(A) i s non w i l l make use n G = ± trivial and is = X, Q A i s elementary A b e l i a n . i s a group and G One n these are a l l . F(A)/cj>(A) = A/<J)(A) we n groups are i n elementary a b e l i a n groups. group, t h e n X <_ QS (P°0 i s an of the f o l l o w i n g non i = 1,2,•••,n. n showing X (1 so a r e A. the Abelian A i s not i n a p o s i t i v e integer define, G., QS Hence, i f A is were i n A n o t h e r l i n e of i n v e s t i g a t i o n i s the c l o s u r e . o p e r a t i o n a p p l i e d to of is X. standard nG = Let Let If q K be a group i n be a prime such t h a t B = qK q j p - 1. and ~B = qK/qK', (1.2.11) we can form then T = LC normal e c c e n t r i c subgroup of p o s i t i v e i n t e g e r such t h a t and suppose A Consider A = BC i n pQ q |K| = p, . q where A = K wr. Cq = C By Pardoes L = L/qK' c o n t a i n e d i n B. q [ p^* - 1. where p wr. Cq. construction i s a minimal Let b By (1.1.3), K = K/K'• be t h e s m a l l e s t |L| = p^. Since qK'<l T e P^, by (1.2.10), i t has a complement H i n T. L e t L* be the complement o f We note t h a t qK' i n L. i . e . L* = H H L. L* = L , and t h e f o l l o w i n g h o l d s . (2.3.7) Lemma : There e x i s t s a subgroup (a) |W*| (b) [ K , W*] = 1. Proof : Consider BCq = Cp wr. Cq . q of order p^. and C p p 1 3 = <x^| x? = 1>, a 2 ) q 3 L* such t h a t 1 o f p-elements. a l 2 a : ( x , • • • ,x Let of - . We can c h a r a c t e r i s e over the f i e l d V = W* B B i s an elementary A b e l i a n as a q - d i m e n s i o n a l v e c t o r space F o r i f {u^,'-',Uq} i = 1,•••,q, v > I i=l p-group a u ± i s a suitable basis f o r then the isomorphism i s g i v e n by, i ; 1 <_ a ± <_ p, i = 1,2, • • • ,q . be t h e map, q 3 : R e s t r i c t t h e map a to L - L I 1=1 a.u. V > a q . and c o n s i d e r t h e map 3a on L*. The l {(x^ a k e r n e l o f t h i s map i s q a dim a (L*) )| a = a q W" of (y^'-y^,!) (2.3.8) Proof : show m-jCp BC L* of order Now x K q a ( L * ) i n which a l l elements as a subgroup o f p^ \ ) | q A n X <_ QS (pQ) t h i s means t h e r e i s a k q £ K } , q so c l e a r l y [W*,K ]=l. q . n (n t i m e s ) . i s i n QS (pQ) r L i n which e v e r y element i s o f t h e form = { ( 1 , k x m Cp r = 1, = BC . q n unique complemented B. where Then i t i s s u f f i c i e n t t o PpP2» * ** »P a r e r distinct a mCp. Choose a prime W = AC series, - 1. Then . q Then p groups w i t h r - a > m p QS (pQ) n p^»P2»* *'>P r i s i n pQ Let a and l o o k as i t has a be t h e s m a l l e s t by the c h o i c e o f q. Also So mC e S (P°0 . p n c o n t a i n s a l l elementary A.belian l d i s t i n c t prime d i v i s o r s . d i s t i n c t prime d i v i s o r s W 1<JA<3W. by (1.1.3), and mC sn a C <l.W. Thus we may assume q > p be a minimal normal e c c e n t r i c subgroup o f Form chief Ip p consider Let A q contained i n A = aC of When do t h i s by i n d u c t i o n on r . Cp wr. C q b - 1 L e t nCp = Cp x ••• x Cp, When at • L* Theorem : primes. and = p (=0). When c o n s i d e r i n g subgroup = p} dim ga(L*) + dim (ker ga) Thus t h e r e i s a subspace o f dimension have q Now c o n s i d e r t h e case o f We c a n o r d e r t h e s e such t h a t r P ^2. r G = m^Cp^ Let G-L = A/B Let r For, i f K maximal normal subgroup o f normal one. In t h i s l p 1 x m r-l P _i K' as * C r can always choose A = K then * *'' c A, (or K), i s the unique and thus must be c y c l i c r = 1 K/K' = K Let We m and a subnormal subgroup passes t h r o u g h a G-^ e case Then as i n the c a s e f o r K', A G^ = and p B <3 A sn K e P Q . where, A <_ K ' . such t h a t x ••• x m C K a o f prime o r d e r . A <_ K ! can be choosen such t h a t be a c y c l i c group o f o r d e r p f o r some prime p. CO As p 1 TT 2 r we c a n use D i r i c h l e t ' s result t h a t the sequence {2 + np„} .. n=I r c o n t a i n s an i n f i n i t e number o f primes r . Thus i t i s p o s s i b l e t o choose a prime r s u c h t h a t K wr. C Consider = C C r r = C wr. C . p c o n s t r u c t i o n t o form a group, normal e c c e n t r i c consider we subgroup o f L wr. = DC^ can form a group H = SC L = R C CC = C As r r^p - 1 e pQ, where . S we where contained i n r wr. Cn r p r p^r - 1 r-fp - 1. can use Pardoe's R C. and i s a minimal Let L = L/L' A g a i n , by Pardoe's i s a m i n i m a l normal and construction eccentric r subgroup o f DCp F contained i n D. Since p | r - 1, r s H wr. be a p r i m e such t h a t C = EC s choose and form T S = C p wr. C . s s > p^ . Let H = H/H', = TC S . The group As J s \ p and pQ. consider By Pardoe's c o n s t r u c t i o n once more we a m i n i m a l normal e c c e n t r i c subgroup o f J H. e _ m +1 Let we have r r - 1 then J e P^. EC S We can contained i n claim has been c o n s t r u c t e d i n the f o l l o w i n g E G s Q S (J). N steps. T J=TC J , S s < J'=T J ' c sH 1 bs i L' P, sp q? K 4 L' asp rsp K' r - B By r|p - 1, p | r a - 1, s|p£ - 1, b r where SH' (1.1.3) the i n t e g e r s ILj = H' . which, by [V,Hg] = 1. Now there a,b,c are the smallest respectively. Let SH' = R| x ... x H exists a subgroup U o f T (2.3.7), has a subgroup V, such t h a t By the c h o i c e of s, c > m, so we may r such t h a t that |v| g complements = p£ - 1 and choose a subgroup m W of V such t h a t (Hj_ x ... x H _ g 1 |w| = p x A).W r r where and [W,H ] = 1. S A sn K' = KjL . C o n s i d e r the subgroup Then, , (H^ x • • • x H _ g x A). W 1 sn sn sn Also (ILj x ... x H _ g centralises H s (Hj_ x . .. x H l ^ x HI)W J x B) <J (Hj_ x ... x 1 x IV _ S ± x A)W H^ x ... x H^_ 1 x A)W a as B <3 A and _ xB = a s . Finally, (Hi x where (H| x ... x H _^ x Kl)W m^x ... x C i s given by, a[(h ,«'-,h _ ,a)w.Hj_ x ... x H _^ * B] 1 s 1 s = (aB, wH[ x . • • x H ) . s This completes the proof. m r P r . CONCLUSION We conclude with some unanswered questions that follow from the above Chapter. First i s QS PQ = X ? It would seem that the answer to this i s n 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 H/K of G. If we can show the holomorph of Cy V = QS V y and i s a chief factor of G C we have G/Cg(H/K) n e L for a l l chief factors then we are done. For pQ <_ V and i s contained i n X — V . K<3 G e Let H/L i.e. satisfy G / K H/K/L/K be a chief factor of G/K. Then and as (H/K/L/K) = C (H/L)/K G QV = V. Thus we need only show SV = y n to complete the proof of the existence of the counter example. This is at the moment unsettled. Finally, given that the f i r s t question i s settled, we may then ask is (Q, S , M, R }P = X ? Q n 0 BIBLIOGRAPHY 1. A l p e r i n J.L. : [1] The c o n s t r u c t i o n and c h a r a c t e r i s a t i o n o f Some C l a s s e s o f F i n i t e Groups. 2. B l e s s e n o h l and Gaschlitz : A r c h i v . der Math. 15, 349-354 (1967). [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. C a r t e r R.W. [1] N i l p o t e n t S e l f N o r m a l i s i n g Groups. 4. Math. Z. 75, 136-139 C a r t e r , F i s c h e r , Hawkes : Groups. J . Algebra [1] Subgroups o f S o l u b l e (1961). Extreme C l a s s e s o f F i n i t e 9, 285-313 (1968). Lemma [1] F r e n c h T r a n s l a t i o n . (2.6). 5. D i r i c h l e t G.C. 6. F i s c h e r , Gaschutz and H a r t l e y [1] I n j e k t o r e n E n d l i c h e r A u f l o s b a r e r Gruppen. Math. Z. 102, 337-339 (1967). 7. Gaschutz W : [1] ' 190, 93-107 8. H a l l P. Zur E r w e i t e r u n g s t h e o r i e de Math. 9, 245-269 (1952). A C h a r a c t e r i s t i c Property 188-200 (1928). o f S o l u b l e Groups J.L.M.S. 12, (1937). [3] Complemented Groups. J.L.M.S. 12, 201-204, [4] On t h e Sylow System o f a S o l u b l e Group . P.L.M.S. 43, 316-323, Huppert B : (1844). E n d l i c h e r Gruppen. J . Math. [1] A Note on S o l u b l e Groups. J.L.MS. 3, 98-105 [2] 9. Jour, Soluble (1937). (1937). E n d l i c h e Gruppen 1., S p r i n g e r - V e r l a g , [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. V I , §7, Formationen, [6] Chap. V I , §11, §12, §13, (726-760). Band 134 (1967). (696-711). ' 10. [7] Chap. VI, 13.7, (747-749). [8] Chap. I, 18.6, (131). Maclain D.H. Groups. 11. Pardoe K. [1] The Existence of Subgroup of Given Order in Finite Proc. Camb. Phil. Soc. 53, 278-285 (1957). [1] An Embedding Theorem in Finite Soluble Groups. Thesis, University of Warwick. (1970). Masters
- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- Embedding theorems in finite soluble groups
Open Collections
UBC Theses and Dissertations
Featured Collection
UBC Theses and Dissertations
Embedding theorems in finite soluble groups Hughes, Peter Walter 1971
pdf
Page Metadata
Item Metadata
Title | Embedding theorems in finite soluble groups |
Creator |
Hughes, Peter Walter |
Publisher | University of British Columbia |
Date Issued | 1971 |
Description | 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[symbol omitted], those with a unique complemented chief series, is all 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[symbol omitted]. The purpose of this is to find a collection of such closure operations whose join applied to P[symbol omitted] is X . In the course of this investigation we introduce a new closure operation M defined by;
MY = { G | G = |
Subject |
Groups -- Theory of |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-05-10 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
IsShownAt | 10.14288/1.0080468 |
URI | http://hdl.handle.net/2429/34390 |
Degree |
Master of Arts - MA |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
Download
- Media
- 831-UBC_1971_A8 H83.pdf [ 2.27MB ]
- Metadata
- JSON: 831-1.0080468.json
- JSON-LD: 831-1.0080468-ld.json
- RDF/XML (Pretty): 831-1.0080468-rdf.xml
- RDF/JSON: 831-1.0080468-rdf.json
- Turtle: 831-1.0080468-turtle.txt
- N-Triples: 831-1.0080468-rdf-ntriples.txt
- Original Record: 831-1.0080468-source.json
- Full Text
- 831-1.0080468-fulltext.txt
- Citation
- 831-1.0080468.ris
Full Text
Cite
Citation Scheme:
Usage Statistics
Share
Embed
Customize your widget with the following options, then copy and paste the code below into the HTML
of your page to embed this item in your website.
<div id="ubcOpenCollectionsWidgetDisplay">
<script id="ubcOpenCollectionsWidget"
src="{[{embed.src}]}"
data-item="{[{embed.item}]}"
data-collection="{[{embed.collection}]}"
data-metadata="{[{embed.showMetadata}]}"
data-width="{[{embed.width}]}"
async >
</script>
</div>
Our image viewer uses the IIIF 2.0 standard.
To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0080468/manifest