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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 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  

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