SUBNORMAL STRUCTURE IN INFINITE SOLUBLE GROUPS
by
D . J . M c C a u g h a n
A t h e s i s submitted for the degree of Doctor of P h i l o s o p h y
in the
The work contained in the body of this thesis is my pwn except where otherwise indicated.
As regards the attached paper [A]: Sections 1 and 2 are
preliminary; Lemmas 3.1-3.5 are due to me; Lemmas 3.6 and 3.7, and the corollary to Theorem B, are due to my co-author; the remainder is joint work not attributable to either author alone.
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A C K N O W L E D G E M E N T S
F i r s t l y I extend m y w a r m e s t t h a n k s to m y s u p e r v i s o r , Dr David M c D o u g a l l , for h i s tolerant and c o n s t r u c t i v e guidance d u r i n g the c o u r s e of t h i s r e s e a r c h .
I also w i s h to thank all t h o s e p e o p l e at the Australian N a t i o n a l U n i v e r s i t y w i t h whom I h a v e w o r k e d , w i t h s p e c i a l thanks to Dr Roger Bryant for his h e l p f u l comments during t h e w r i t i n g of this t h e s i s .
I thank Mrs Barbara Geary for her p a t i e n c e and her s k i l f u l t y p i n g .
SUMMARY
T h e b a s i c c o n c e p t s of this thesis are t h o s e of s u b n o r m a l s u b g r o u p and s u b n o r m a l i n d e x . Our aim is to investigate t h e p r o p e r t i e s of groups in some classes defined b y subnormality c o n d i t i o n s , for
e x a m p l e t h e class of groups in w h i c h the s u b n o r m a l indices are b o u n d e d . A s s o c i a t e d w i t h this are two larger and distinct c l a s s e s , the first c o n s i s t i n g of t h o s e groups w h i c h h a v e the s u b n o r m a l intersection
p r o p e r t y , that i s , in w h i c h the intersection of any family of s u b n o r m a l subgroups is a g a i n s u b n o r m a l , the second consisting of those groups w h i c h h a v e the s u b n o r m a l join p r o p e r t y , defined a n a l o g o u s l y .
We a t t e m p t to answer two g e n e r a l q u e s t i o n s ,
(i) U n d e r w h a t r e s t r i c t i o n s w i l l a soluble group w i t h some s u b n o r m a l i t y condition b e nilpotent?
(ii) U n d e r w h a t r e s t r i c t i o n s w i l l a soluble group w i t h the s u b n o r m a l intersection p r o p e r t y have a bound on its s u b n o r m a l indices?
After an introductory c h a p t e r , the d e f i n i t i o n s and properties of the v a r i o u s classes are treated in C h a p t e r 2 . In Chapter 3 w e present some t e c h n i c a l r e s u l t s to b e used in later i n v e s t i g a t i o n s . C h a p t e r 4 deals w i t h the t o p i c of "rank" in soluble groups and leads up to the r e s u l t that an extension of a group with the m i n i m a l condition on s u b n o r m a l s u b g r o u p s extended b y a group w i t h b o u n d e d s u b n o r m a l indices a g a i n h a s b o u n d e d s u b n o r m a l i n d i c e s .
generalisation is elusive.
In Chapter 6, using a seemingly new restriction, two further results of type (i) are proved. The first of these is used to show that a soluble minimax group with the subnormal intersection property has a bound on its subnormal indices. An example is constructed to show that the same is not true for soluble groups of finite reduced rank.
Also included in the thesis is the joint paper [A] which applies some earlier results to investigate and characterise
ACKNOWLEDGEMENTS SUMMARY
CHAPTER 1 : CHAPTER 2 :
CHAPTER 3 :
CHAPTER 4 :
CHAPTER 5 :
CHAPTER 6 :
Introduction
Subnormal subgroups 2.1 Preliminaries 2.2 Subnormal subgroups
2.3 Bounds on subnormal indices
2.4 The subnormal intersection property 2.5 Joins of subnormal subgroups
TT-torsion-freeness and 7T-radicability 3.1 Series and central series
3.2 T T - t o r s i o n - f r e e n e s s 3.3 Quasi-TT-radicability 3. 4 TT-radi cabi lity
3.5 Five easy lemmas
Some finiteness conditions for soluble groups 4.1 Rank
4.2 Finiteness conditions for abelian groups 4.3 Finiteness conditions for soluble groups 4.4 Some results
Some results on subnormal structure 5.1 Abelian-by-cyclic groups
5.2 Abelian-by-nilpotent groups 5.3 Metanilpotent groups
5.4 Abelian-by-finite groups
Subnormal structure in soluble groups of finite rank
6.1 A useful theorem
6.2 Soluble minimax groups with the subnormal intersection property
6.3 An e xample REFERENCES
Attached PAPER [A]
INTRODUCTION
It w o u l d b e a fair statement to say that the c e n t r a l theme o f this t h e s i s o r i g i n a t e s from the concept of "nachinvariant" subgroup introduced b y W i e l a n d t in 1939 ([3?])- His generalisation of the n o r m a l i t y of a subgroup w a s later termed "accessibility" and finally
" s u b n o r m a l i t y " , the latter being standard usage t o d a y . At first the d e v e l o p m e n t of W i e l a n d t ' s idea was confined principally to the t h e o r y of finite g r o u p s , but in the last ten y e a r s , m a i n l y due to the efforts of Robinson and R o s e b l a d e , its a p p l i c a t i o n s to the study of infinite g r o u p s have b e e n explored to some e x t e n t .
From this s t a n d p o i n t , a group is considered in terms of its " s u b n o r m a l s t r u c t u r e " , a term which includes such questions as how m a n y s u b n o r m a l subgroups there a r e , where t h e y fit into the lattice of a l l subgroups of the g r o u p , and whether or not they are
w e l l - b e h a v e d in some sense or o t h e r . Investigations from this point of view have g e n e r a l l y been directed towards soluble g r o u p s , for in this r e s p e c t simple groups live up to their n a m e b y having t r i v i a l s u b n o r m a l s t r u c t u r e . At the o p p o s i t e end o f the spectrum are n i l p o t e n t g r o u p s , for in these each subgroup is s u b n o r m a l . The i n t e r p l a y b e t w e e n the subnormal structure and the g e n e r a l structure of g r o u p s lying b e t w e e n these two extremes has r e c e i v e d attention in r e c e n t y e a r s . In p a r t i c u l a r , interest seems to have centred on
c o n d i t i o n s on g r o u p s t r u c t u r e , b o t h s u b n o r m a l and g e n e r a l , which w i l l b r i n g the g r o u p "close" to n i l p o t e n c y in some w a y . Many of the
and it is with these that we begin chapter 2. The idea of subnormal subgroup is made precise: in particular, Robinson's useful concept of standard series is introduced and developed. The question of "where" a subnormal subgroup lies is embodied in the notion of defect or
subnormal index. Although most of the material of this chapter is due to Robinson and can be found in his papers [22] - [2S] or in [29], I felt it advisable to include the majority of proofs, not only for their intrinsic interest, but also to facilitate the transition to later chapters, where some of the techniques will be used continually and often without explicit reference.
In later stages of the chapter I introduce some classes of groups which are crucial to the topics discussed in the thesis. The first of these is the class of groups in which the subnormal Indices are
bounded (so that in this sense the subnormal subgroups are
well-behaved). This class is the union of an ascending chain of proper subclasses, the first of which consists of those groups in which every subnormal subgroup is normal. Groups with this property, that is, in which "normality is transitive", have been investigated quite intensely, in the context of both finite groups ([2], [7], [2S],
intersection property. This is the other important class discussed in chapter 2. The bulk of this thesis consists of attempts to obtain information about soluble groups in this class, beginning, usually, in very simple situations and then seeking to generalise or extend the results to more complex ones, following the pattern of [24]. Such
information is contained in chapters 5 and 6, as well as in the attached paper [A].
In dealing with groups which have the subnormal intersection property, it transpires that the notions of TT-radicability and TT-torsion-freeness play an important part. Chapter 3, therefore, begins with an outline of some basic facts on these concepts, which fits conveniently into the setting of Zi4-groups. These properties have been studied in [3], [5] and [I^]. Many of the proofs can be found in [J 7] or [29], so they are generally omitted. The most
important part of Chapter 3, as regards applicability for the purposes of this thesis, is Section 3.5, in which are proved a series of
inter-related lemmas. They deal with the action of automorphisms on abelian groups which are restricted in terms of TT-radicability or 7T-torsion-freeness. These simple and easily proved results do not seem to appear anywhere in the literature, perhaps because of their specialised nature. Some of them may have wider applications.
In Chapter 4 I have attempted to give a fairly brief and cohesive account of some aspects of the thorny topic of "rank" in soluble
Chapter 6, where some knowledge of soluble groups of finite rank is essential background. So also is a familiarity with soluble minimax groups, further information on which may be found in [/] and [27].
Chapter 5 sees the first use of the lemmas of Chapter 3 to prove results linking subnormal structure with nilpotency. An important example of this type of result is Lemma 4 of [24], in which Robinson shows that a group with the subnormal intersection property which is a cyclic extension of a free abelian group of finite rank is nilpotent, Motivated by this result, but more interested in abelian-by-finite groups (for reasons which will become clear in the discussion of chapter 6), I proved the fundamental theorem 5.11, at first in a very complex manner, but then more simply as the essential features become evident. Most of the remainder of chapter 5 investigates simple
cases of soluble groups with the subnormal intersection property, from which one can deduce detailed information on more general groups of the same type. Unfortunately the complexity of the general picture means that the task of characterising these groups is likely to prove a difficult one. In spite of this some useful results are obtained.
final theorem 5.46 of this chapter.
In chapter 6, attention focuses on soluble minimax groups. The reason for this interest lies in the following facts: a soluble group with the minimal condition on subgroups has a bound on its subnormal
indices ([25], Lemma 3.2); and a soluble group with the maximal condition on subgroups, being finitely generated, has a bound on its subnormal indices if it has the subnormal intersection property, by a result of Robinson ([24], Theorem A). It is natural to ask, therefore, "For what classes of groups containing all soluble groups with the minimal or maximal conditions on subgroups do the two subnormality properties coincide?" The class of soluble minimax groups springs to mind.
It is at this stage that the reason for my interest in abelian-by-finite or, more generally, nilpotent-abelian-by-finite groups becomes clear, for in this context, McDougall ([20], Theorem A) has proved a result showing that the above question, when restricted to soluble minimax groups, reduces to a consideration of the nilpotent-by-finite case. Once again the key theorem (6.12) is one which deals with
arbitrary nilpotent-by-finite group with the subnormal intersection property has a bound on its subnormal indices. The indications of chapters 5 and 6 are that it should, but the proof will not be easy.
Part of this thesis is the paper [A], which was written jointly in late 1971 with my supervisor. He realised that some of my results (in particular, the one which appears as Lemma 3.5 of [A]) could be used to investigate metanilpotent groups with bounds on the subnormal indices of their subnormal subgroups, his earlier efforts in this direction having foundered for lack of suitable tools.
The initial sections, 1 and 2, are purely preliminary; indeed section 2 is a resume of some of the material covered in chapters 2 and 3 of this thesis. Section 3 begins with a sequence of technical lemmas: 3.1 - 3.5 are due to me, and 3.6 - 3.7 to my co-author. Some of these lemmas have been restated, for convenience, in the body of the thesis, with references to [A] for the proofs. With the
exception of the corollary to Theorem B, which is due to my co-author, the rest of the results of [A] are joint work not attributable to either of us alone.
The main results of the paper concern nilpotent-by-(periodic nilpotent) groups, but are more readily explained for periodic
SUBNORMAL SUBGROUPS
The first section of this c h a p t e r sets out some basic notation and t e r m i n o l o g y , m a i n l y w i t h regard to c o m m u t a t o r s ; two simple lemmas are r e c o r d e d for future u s e . Then w e proceed to the m a i n topic of the c h a p t e r - indeed o f the thesis - b y introducing in Section 2.2 the c o n c e p t o f a s u b n o r m a l s u b g r o u p . After d e v e l o p i n g some of the e l e m e n t a r y t h e o r y of s u b n o r m a l s u b g r o u p s , we d e v o t e the remaining sections of t h e chapter to a d i s c u s s i o n of groups w h i c h are restricted in some w a y b y c o n d i t i o n s on t h e i r s u b n o r m a l s u b g r o u p s . The
r e s t r i c t i o n s treated in Sections 2.3 and 2."4 are c e n t r a l to the s u b j e c t - m a t t e r of the t h e s i s . These a r e , r e s p e c t i v e l y , the
r e q u i r e m e n t that there should be a b o u n d on the defects of subnormal s u b g r o u p s , and the r e q u i r e m e n t that the intersection of any family of s u b n o r m a l subgroups should b e s u b n o r m a l . Section 2 . 5 , w h i c h deals w i t h joins of s u b n o r m a l s u b g r o u p s , though perhaps of independent
i n t e r e s t , is less important and therefore less d e t a i l e d .
2.1 Preliminaries
2.11 D E F I N I T I O N . if {E. i ^ 1} is a collection of subsets %
o f a group G , w e d e n o t e b y i : i € I) the subgroup of G
g e n e r a t e d b y the subsets H^ , i i I , that i s , the smallest subgroup
o f G c o n t a i n i n g t h e m . (Superfluous braces w i l l be o m i t t e d . )
2.12 D E F I N I T I O N . (a) If x and y are elements of a g r o u p , - 1
we d e n o t e b y [x^, x ^ , x ^ the element CC ^ * • • • • CC - 9
- 1 ' n - P n
(b) If Z and Y are subgroups of a g r o u p , we denote by
X^ the s u b g r o u p { •. x ^ X, y i 1) we denote b y [ I , I] the s u b g r o u p {[x, y] : x i X , y € I) , the commutator of X and Y . If X ^ , X ^ , ..., are subgroups of a g r o u p , with n > 2 , we denote
b y [X , X X ^ the subgroup
J. /C yt - 1 ' ' n-l-J n
W e state in the form of a lemma some w e l l - k n o w n and easily v e r i f i a b l e p r o p e r t i e s of c o m m u t a t o r s .
2.13 L E M M A . If x, y and z are elements of a groug then
- 1 (i) {x, y] ^ [y, x]
Hi) (a) [x, yz] ^ [x, s][x, y] = [ x, z][ x, y][ x, y, s] ;
(b) [xy, z] ^ [x, z]^[y, z] ^ [x, z][x, z, y][y, z] ;
(Hi) [x, y z xY [z, x y j ^ = 1 .
We can now deduce some u s e f u l and w e l l - k n o w n r e l a t i o n s between c o m m u t a t o r s u b g r o u p s .
2.14 L E M M A . If X, Y and Z are subgroups of a group, then
(i) [Z, H - [I, I] ;
(Hi) X^ = (X, [X, I]) ^ X[X, Y] ;
(iv) [X, Y, Y] S[X, Y] ;
(V) X'^'^'-X';
(vi) i f Y^ ^ Y then [ Z , ( 7 , Z>] = < [ Z , Z] , [ X , Y] .
P r o o f : (i) and (ii) f o l l o w e a s i l y from 2.13 (i) and (ii)
r e s p e c t i v e l y . (Hi) is i m m e d i a t e from t h e d e f i n i t i o n , u s i n g (i) and (ii) t o o b t a i n t h e s e c o n d e q u a l i t y . (iv) is a c o n s e q u e n c e o f (ii) and
Y
(Hi). To p r o v e (v) , n o t e t h a t { X, Y) - YX , so that
(vi) is i m m e d i a t e from 2.13 (ii).
The f o l l o w i n g r e s u l t , t h e " t h r e e s u b g r o u p l e m m a " , is a c o n s e q u e n c e o f 2.13 (Hi).
2.15 LEMMA {[ S] , T h e o r e m 2 . 3 ) . I f X, Y and Z are normal subgroups of a group G then
[ I , Y , Z] S lY, Z, X][Z, X, Y] .
P r o o f : It is e a s y to see t h a t each o f t h e c o m m u t a t o r s u b g r o u p s i n v o l v e d is n o r m a l in G . It w i l l t h e n s u f f i c e to show t h a t
[X, Y, Z] is t r i v i a l on t h e a s s u m p t i o n t h a t t h e s u b g r o u p
showing that [ X , Y , Z] = 1 , as r e q u i r e d .
In dealing with commutator subgroups, two situations frequently arise for which it is useful to have some concise n o t a t i o n .
2.16 DEFINITION. (a) If Z and I are subgroups of a group
we define yXY for each ordinal a > 0 by
if a is a positive (that i s , non-empty) non-limit ordinal;
e<a
if a is a limit ordinal.
(b) If G is a group we define yGcx for each positive
ordinal a by
y.G ^ G y^G = G] 'a-1
if a is a non-limit o r d i n a l , a > 1 ;
y G = n y G
^ 3<a ^
if a is a limit ordinal.
No confusion should arise between these two notations. There is of course a close connection between them: in fact if a is a finite
positive ordinal y^G = yGG^"'^ , whereas for an infinite ordinal 3 ,
The lower central series of G is the descending chain of fully invariant subgroups of G defined in 2.16 (b). Its second term
y^G = [G, G] = is the derived group of G , and the derived series
of G cap be defined by
if a is a positive, non-limit ordinal;
B<a
if a is a limit ordinal.
We will assume here the elementary properties of soluble and nilpotent groups, that is, groups in which the trivial subgroup appears after finitely many terms of the derived series or lower central series respectively.
We conclude this section with a useful lemma, a variation on Theorem 2 of [ JO] .
2.17 LEMMA. If H is a normal subgroup of a group G then for each positive integer k j
[H, Jj^G'] 2 yHG^ .
Proof: We proceed by induction on k , noting firstly that
Y^G] - [H, G] = yHG^ .
h a v e
H, Y^Cj = LY^G, H
b y d e f i n i t i o n . H e n c e , a p p l y i n g Lemma 2 . 1 5 , we obtain
[H, y^G] 5 [G, H, G] , 'fe-r
since a l l the s u b g r o u p s involved are n o r m a l in G . But by the induction h y p o t h e s i s .
]G, H] , < = yHG^ .
By the same t o k e n .
= yHG^
It f o l l o w s that [H, y-jG'] 5 y H G ^ , which completes the p r o o f .
2.2 Subnormal Subgroups
The b a s i c idea of this section is due to Wielandt ([ 31] ); the t r e a t m e n t l a r g e l y follows that of Robinson [ 29] .
2.21 D E F I N I T I O N . A subgroup 5 of a group G is said to be subnormal in G w h e n there is a chain of subgroups
i such that 0 < i < r . If such a chain e x i s t s , then there w i l l b e one o f m i n i m a l l e n g t h , that i s , a chain in which the number of
n o n - t r i v i a l factors is l e a s t . ' This number is clearly
i n d e p e n d e n t o f the c h o i c e of a m i n i m a l such chain: it is called the subnormal index or defect of E in G , and denoted by siG •. H) .
The f o l l o w i n g r e m a r k s are t r i v i a l consequences of the d e f i n i t i o n .
(a) If H is a s u b n o r m a l subgroup of G and K is any s u b g r o u p of G , then U ^ K is s u b n o r m a l in K , and
s{K •. E K) S s{G : E) .
(b) If K is a s u b n o r m a l subgroup of G , and E is a subnormal subgroup of K , then E is s u b n o r m a l in G and
s{G : E) 5 s{G : K) + s{K : E) .
(c) If E is a s u b n o r m a l subgroup of G , and N is a normal subgroup of G , then EN is s u b n o r m a l in G , EN/N is subnormal in G/N , and
s{G : EN) = s{G/N : EN/N) 2 s{G : E) .
Before p r o c e e d i n g to a m o r e detailed discussion of subnormal s u b g r o u p s , we d i g r e s s to prove a lemma w h i c h , as w e l l as being of use in later c h a p t e r s , serves to show the s i m i l a r i t y , in some
c i r c u m s t a n c e s , of the b e h a v i o u r of n o r m a l and subnormal s u b g r o u p s . It is a g e n e r a l i s a t i o n , due to R o b i n s o n , of a well-known theorem of
F i t t i n g .
subgroup of G . Then G is nilpotent.
Proof: Let H = H ^ ^ ... S E^ - G be a chain of V r-1 0
subgroups as in 2.21. Then H^ is certainly nilpotent. We show
that, for 0 < i •S T , the nilpotency of E. implies that of E.
since E^ - G this will suffice to prove the lemma.
Suppose that E. is nilpotent. Since E.N - G ,
h-1
=
h-1
^ V = ^ ^^ '
by t h e m o d u l a r law.
Both E. and E. N are nilpotent normal subgroups of
E^ ^ , so that, by Fitting's theorem, their product is nilpotent,
and the proof is complete.
Now we return to the main theme to develop an alternative approach to subnormal subgroups.
2.23 D E F I N I T I O N . If E is a subgroup of a group G , the
standard series of E in G is a descending chain of subgroups of
G , each containing E , defined for each ordinal a by
, , .
^G.a __if a is a p o s i t i v e non-limit o r d i n a l ;
if a is a limit ordinal.
We make some remarks on this definition.
(a) For any given group G , there is an ordinal X such that (7 A G X+l
for any subgroup H of G , H ' - H ' . It is then clear that
for any ordinal a > A , i f = H ^ , and the standard series of H in G may be taken as
{E^'"" : a 5 A} .
(b) It is a simple matter to show that if H and K are any subgroups of a group G , and a is an ordinal, then
iH n 5 n
and
(c) If H is a subgroup and N a normal subgroup of G , then for any finite ordinal ot ,
and
where G = G/N and H = HN/N
(at l e a s t for the "early" t e r m s ) , u s i n g the c o m m u t a t o r notation i n t r o d u c e d in 2 . 1 6 .
2.24 L E M M A . Let E be a subgroup of G . Then for any
non-negative integer i ,
= HyGH^ .
Proof: First we r e m a r k that for each i , yGH is normalised b y H , u s i n g 2.14 (ii), so that the r i g h t hand side of the equality
is indeed a s u b g r o u p . We prove the statement b y induction on i ,
n o t i n g that H ^ = G = H y G H ^ b y d e f i n i t i o n . If i is a positive i n t e g e r , and the statement is true for a l l n o n - n e g a t i v e integers less than i , t h e n we h a v e
= h] b y 2.14 (Hi)
- H [ H y G H ^ by inductive h y p o t h e s i s
= , h\ by 2.14 (vi) .
Hence = H y G H ^ , w h i c h completes the p r o o f .
We now prove a lemma linking the idea of subnormality w i t h both v e r s i o n s of the standard s e r i e s .
2.25 L E M M A . The following three conditions on a subgroup H of
a group G are equivalent:
(ii) H = H^'^ ;
(iii) yGlf 5 H .
Proof: Suppose (i) is true. Then there is a chain
r r-1 0
of subgroups, each normal in the next. We prove by induction that for
each i with 0 S i S r , E^'^ S E^ , noting that E^'^ = G = E^ . •
Suppose that Q < i S v and that E ^ S . Then
= / < fl ^ 5 ^ = fl. .
G r
The inductive proof is complete, and E " - E follows on taking i - V , proving that (i) implies Hi).
The equivalence of (ii) and (Hi) is immediate from Lemma 2.24.
Now suppose that {ii) is true. By definition 2.23, each term E of the standard series of E is normal in its predecessor E
for each positive integer i . The chain
is then precisely of the kind described in 2.21, and has at most r non-trivial factors. Thus E -is subnormal in G with
siG : E) S r , establishing (i) and completing the proof of the lemma.
IS if E ^ " ^ < ^ for some p o s i t i v e integer r , then E ^ s u b n o r m a l in G w i t h defect p r e c i s e l y r .
2.3 Bounds on subnormal indices
In this section we consider some b a s i c properties of the class of g r o u p s in w h i c h the s u b n o r m a l indices of s u b n o r m a l subgroups are
b o u n d e d . By a class of groups we m e a n a class in the u s u a l sense which c o n t a i n s a t r i v i a l g r o u p , and w i t h every group in the c l a s s , all its
i s o m o r p h i c c o p i e s . The class m e n t i o n e d m a y be considered as the union of an a s c e n d i n g chain of proper s u b c l a s s e s , one for each non-negative integer n . The subclass c o r r e s p o n d i n g to n is composed of those g r o u p s in w h i c h the d e f e c t s of the s u b n o r m a l subgroups do not exceed n . For e x a m p l e , when n = 0 the subclass contains only t r i v i a l
g r o u p s , and w h e n n - 1 each group in the subclass has the property that any s u b n o r m a l subgroup is n o r m a l - "normality is t r a n s i t i v e " .
Our first lemma c h a r a c t e r i s e s groups in any of these subclasses in t e r m s of the standard series of an arbitrary s u b g r o u p .
2.31 LEMMA ([24] , Lemma 2 (ii)). A group G has a hound n
on its subnormal indices if and only if, for each subgroup E of G , ^G,n ^ ^G,n+r ^ y^i^^^Q p is any non-negative integer.
fl ,n „G,n+r ^ Proof: If G has a subgroup E such that E > E , ror some p o s i t i v e integer r , then it is clear from the definition of
standard series that > • But b y the r e m a r k following
On the o t h e r h a n d , if for each subgroup H of G ' , H ^ = for any n o n - n e g a t i v e integer r , this w i l l hold a fortiori when E is a s u b n o r m a l s u b g r o u p . Since E c o i n c i d e s w i t h some term of its standard s e r i e s , b y Lemma 2 . 2 5 , it follows t\iat H - H^ , w h i c h , b y the same l e m m a , shows s ( G : E) S n . Now E was an arbitrary
s u b n o r m a l s u b g r o u p of G , so the proof of the lemma is c o m p l e t e .
The f o l l o w i n g p r o p o s i t i o n is an immediate consequence of r e m a r k s (a) and (c) a f t e r D e f i n i t i o n 2 . 2 1 .
2.32 P R O P O S I T I O N . Let K be a subnormal subgroup and N a normal subgroup of a group G which has a bound on its subnormal indices. Then K and G/N have the same bound on their subnormal indices.
(We say that s u b n o r m a l subgroups and quotient groups "inherit" the p r o p e r t y of h a v i n g a bound on the subnormal indices of their subnormal s u b g r o u p s . )
In his s e m i n a l paper [ 3/] , Wielandt was concerned largely with groups h a v i n g a c o m p o s i t i o n series of finite length (see [ 16] , p . 1 1 2 ) . The t e r m s of such a s e r i e s , if it e x i s t s , are all subnormal subgroups of the g r o u p ; indeed the following l e m m a , which is an obvious
c o n s e q u e n c e of the J o r d a n - H o l d e r - S c h r e i e r t h e o r e m , shows that the l e n g t h of such a series gives a bound for the defect of any subnormal s u b g r o u p .
2.33 L E M M A . If G is a group with a composition series of
of G has at most n + 1 members.
We can u s e t h i s to d e r i v e a m o r e g e n e r a l r e s u l t .
2.34 LEMMA ([24] , Lemma 1 ) . Let G be a group with a normal subgroup N which has a composition series of length n , If G/N has a bound r for its subnormal indices, then G has the bound
r + n for its subnormal indices.
Proof: Let H be any subgroup of G . Because G/N has the b o u n d r for the d e f e c t s of its s u b n o r m a l s u b g r o u p s , we may apply
Lemma 2 . 3 1 to the standard series of EE/N in GJ/N . Bearing in mind r e m a r k 2.23 ( c ) , w e d e d u c e t h a t , for any n o n - n e g a t i v e integer t ,
Now the d e s c e n d i n g chain
n iv > n / ! / > . . . > n E
of s u b n o r m a l subgroups of N has n + 2 m e m b e r s . By Lemma 2 . 3 3 , t h e r e f o r e , there is an integer k with r < k S r+n such that
n N -- E^^^'"- n N .
Since E ^ ' ^ N = E ^ , it follows that
This m e a n s t h a t , at w o r s t ,
proving the result in view of Lemma 2.31.
In most applications of this r e s u l t , N is a finite g r o u p . If we transpose the conditions on N and G/N in Lemma 2 . 3 4 , the
conclusion of the lemma does not hold: it is not true that a finite extension of a group with bounded subnormal indices again has bounded subnormal indices.
2.35 EXAMPLE. Consider the infinite dihedral group
D ^ ia, h : iab)^ = b^ - 1) .
The chain of subgroups {D^ : n > 0} defined by
is the standard series of the subgroup B - ib) , because D^ - D ,
and for each integer n > 0 ,
D
B^ ^ B[D , S
n •
^{b, b ) =
2nt
s a
- ^ n f l
Now for each n > 0 , < D ^ ; thus by the remark after
Lemma 2.25 we have s[D : dJ = n , so that D has unbounded
subnormal indices. This is in spite of the fact that D has a normal abeli^n subgroup A =( a) with D/A a 2-cycle.
s u b n o r m a l i n d i c e s : in p a r t i c u l a r any n i l p o t e n t group w i l l have this p r o p e r t y . For if G is n i l p o t e n t of class c and H is any
s u b g r o u p o f G ,
yGE^ S y^^^G - 1 .
By 2.25 w e see that H is s u b n o r m a l in G with subnormal index at m o s t a .
Great interest centred for some years on converses to this
s i t u a t i o n . If a group has every subgroup s u b n o r m a l , is it n e c e s s a r i l y n i l p o t e n t ? If a group has every subgroup s u b n o r m a l of bounded d e f e c t , is it n e c e s s a r i l y n i l p o t e n t ? The first question was answered in the n e g a t i v e b y H e i n e k e n and M o h a m e d , who in 1968 ([14]) c o n s t r u c t e d , for each prime p , a m e t a b e l i a n p - g r o u p which is not n i l p o t e n t although e v e r y p r o p e r subgroup is n i l p o t e n t and s u b n o r m a l .
The second question was given a p o s i t i v e answer by Roseblade in some v e r y deep work in 1965 ([ 30] ). We can state the r e l e v a n t result as f o l l o w s :
2.36 THEOREM ([30] , Corollary to Theorem 1 ) . There is a function R on the set of positive integers such that if G is a group in whioh every subgroup is subnormal of defeat at most n , then G is nilpotent of class at most R(n) .
An immediate c o n s e q u e n c e is the f o l l o w i n g r e s u l t , w h i c h does not seem to a p p e a r in the l i t e r a t u r e .
indices, then the lowev oentval series of G terminates after a finite nxmber of steps; indeed there is a non-negative integer j
depending only on the bound, such that for any non-negative integer k
.
y .G = Y- . J J+fc
Proof: Let n be the bound on the defects of subnormal
subgroups of G . If n = 0 there is nothing to prove. If n > 0 , choose j = Bin) . Then for any fe > 0 , is n i l p o t e n t , thus
has every subgroup subnormal. B u t , by Proposition 2 . 3 2 , the bound n on subnormal indices is inherited by the factor group ' which
therefore must have nilpotent class at most j , by Theorem 2.36. It follows that y .G S y . yG , a n d , since the reverse inclusion is
t r i v i a l , the corollary is proved.
Although 2.37 proves useful later in the t h e s i s , perhaps a more interesting consequence of Roseblade's theorem is the following unpublished result of R o b i n s o n .
2.38 THEOREM (Robinson). The direct product of two groups,
each with a bound on its subnormal indices, again has a bound on its subnormal indices.
Proof: Suppose G ^ M ^ N where M and N have a bound k for the subnormal indices of their subnormal subgroups. Let H be any subnormal subgroup of G . Write
Note that since M ^ H is normalised by E and by H , M^ is
normal in M^ . Similarly is normal in N^ . Note also that
Wq X S 5 < X /i/^ .
Now we prove that, for any positive integer i ,
1 1 i-l
We use induction on i , remarking that for i - l the statement is trivial. Suppose that i > 1 and that
i-2
Then
yM^H i-l yM^H^ H
5 Y- J M '><N using the induction hypothesis 'Z'-L _L -L _L
yM^E^ ^, M n M
yM^H'^ m
yM^H^"^, H since ^M^, NJ =1 .
i-l
Thus y.M - yM E , as required. ly J- -L
Now because E is subnormal in G , there is an integer m such
M^/MQ is n i l p o t e n t . But M ^ is s u b n o r m a l in M , b y r e m a r k s (c) and
(a) a f t e r 2 . 2 1 , so M ^ / M ^ inherits the bound k for its subnormal
i n d i c e s . By T h e o r e m 2 . 3 6 , the n i l p o t e n t class of ^^/^Q most
V ^ R { k ) , h e n c e y M ^ l f < M ^ .
B y an e x a c t l y p a r a l l e l a r g u m e n t , Y ^ ^ ^ - ^'Q • Now it is clear
that for i > 0 , x tl-^Il^ is the direct product of yM^''^ and
yN^H^ ; h e n c e x E^lf S H , that i s , s {M^ x N^ : h] S r . But
M^ h a s defect at m o s t k in M , hence in G , and similarly N^
has d e f e c t at m o s t k in G . We deduce that s [G : M^ N^] S k ,
and f i n a l l y , by r e m a r k (b) after 2 . 2 1 , that ?iG : H) S k "t r . Since this integer is independent of the choice of H , the theorem is
p r o v e d .
It w o u l d b e interesting to h a v e an "elementary" proof of this r e s u l t ; the h e a v y m a c h i n e r y used seems i n a p p r o p r i a t e .
We can n o w prove a m o r e g e n e r a l result in the same v e i n .
2.39 T H E O R E M . Let M^ and M^ be normal subgroups of G with
G/M^M^ nilpotent. If G/M^ and G/M^ have a bound for their
subnormal indices, then so does G/M^ n M^ .
Proof: There is no loss of g e n e r a l i t y in assuming n = 1 .
The map 6 defined by
gQ = [gM^, gM^]
is a monomorphism from G into G/M^ x G/M^ . Write
N^ = {[m^M^, M^ : m^ € M^]
and
Then and N ^ are subgroups of GG ; s o , t h e n , is their
product
Now N is normal in G / M ^ x G/M^ , and the corresponding factor
group is isomorphic to G / M ^ M ^ x G / M ^ ^ , so is nilpotent. It follows
that GQ is a subnormal subgroup of G/M^ x g /M^ . This latter
g r o u p , h o w e v e r , has a bound on its subnormal indices, by Theorem 2.38; thus , and its isomorphic copy G , must have a bound for its subnormal i n d i c e s , as r e q u i r e d .
2.4 The Subnormal Intersection Property
In this section we introduce another condition on subnormal
subgroups which w i l l prove to be less restrictive then that of Section 2.3.
t h a t in a n y g r o u p t h e i n t e r s e c t i o n o f t w o s u b n o r m a l s u b g r o u p s is a s u b n o r m a l s u b g r o u p . It is n o t n e c e s s a r i l y t h e c a s e , h o w e v e r , t h a t t h e i n t e r s e c t i o n o f an a r b i t r a r y f a m i l y o f s u b n o r m a l s u b g r o u p s is
s u b n o r m a l . For e x a m p l e , in t h e i n f i n i t e d i h e d r a l g r o u p , w i t h t h e n o t a t i o n of 2 . 3 5 , t h e c h a i n o f s u b n o r m a l s u b g r o u p s [d : n > O}
yx
i n t e r s e c t s in t h e s u b g r o u p B . But B c a n n o t b e a s u b n o r m a l s u b g r o u p , b y L e m m a 2 . 2 5 , s i n c e its s t a n d a r d series
= P : n > 0 n
d o e s n o t b e c o m e s t a t i o n a r y .
We s a y t h a t a g r o u p G h a s t h e subnormal intevseotion property if t h e i n t e r s e c t i o n o f an a r b i t r a r y f a m i l y of s u b n o r m a l s u b g r o u p s of G is a g a i n a s u b n o r m a l s u b g r o u p of G . The i n f i n i t e d i h e d r a l g r o u p , t h e r e f o r e , f a i l s t o h a v e t h e s u b n o r m a l i n t e r s e c t i o n p r o p e r t y . That it is in s o m e s e n s e t y p i c a l o f such g r o u p s w i l l b e seen from t h e f o l l o w i n g l e m m a .
2.41 L E M M A ([24] , Lemma 2 ( i ) ) . A group G has the subnormal intersection properly if and only if the standard series of every subgroup becomes stationary after finitely many terms.
P r o o f : S u p p o s e G h a s t h e s u b n o r m a l i n t e r s e c t i o n p r o p e r t y . Let H b e a n y s u b g r o u p , and let
Y = : i > 0}
Because H is contained in Y , each term of the standard series of H is contained in the corresponding term of the standard series of y , by remark 2.23 (b). But if d is the defect of I in , this means that
hence for any integer i > 0 ,
This proves one half of the lemma, since H was an arbitrary subgroup. Conversely, let
H = n{ff. : i € J} %
be the intersection of an arbitrary family of subnormal subgroups of G . Then, by assumption, the standard series of ff in G becomes stationary, at E , say, after finitely many terms, ff is a subnormal subgroup, by Lemma 2.25 and the succeeding remark. Now for each i in I , i?. , being a subnormal subgroup, coincides with a term of the
standard series of E. in G , and therefore contains the corresponding
term of the standard series of E in G , by remark 2.23 (b). A fortiori, each E^ certainly contains E . It follows easily that
and that E is subnormal in G , completing the proof of the lemma.
2.42 P R O P O S I T I O N . Let K be a subnormal subgroup and N a normal subgroup of a group G which has the subnormal intersection
property. Then K and G/N have the same property.
It m a y b e o f i n t e r e s t to n o t e an a l t e r n a t i v e c h a r a c t e r i s a t i o n of g r o u p s w i t h t h e s u b n o r m a l i n t e r s e c t i o n p r o p e r t y , w h i c h d o e s not seem t o a p p e a r in t h e l i t e r a t u r e .
2.43 L E M M A . A group G has the subnormal intersection property
if and only if for each subgroup H of G the family of all subnormal
subgroups of G containing H has a unique minimal member E .
P r o o f : If G h a s the s u b n o r m a l i n t e r s e c t i o n p r o p e r t y , we can t a k e H to b e t h e i n t e r s e c t i o n of a l l s u b n o r m a l s u b g r o u p s o f G c o n t a i n i n g H . C o n v e r s e l y , if G s a t i s f i e s t h e second c o n d i t i o n and H is a n y s u b g r o u p of G , t h e c o r r e s p o n d i n g H is c l e a r l y contained
in e v e r y t e r m o f t h e s t a n d a r d series of H in G . But E c o i n c i d e s w i t h t h e k - t h t e r m o f its own standard s e r i e s , for some n o n - n e g a t i v e
i n t e g e r k , h e n c e E c o n t a i n s t h e c o r r e s p o n d i n g term of t h e s t a n d a r d s e r i e s o f E . T h u s E is the t e r m i n a l point of t h e
s t a n d a r d s e r i e s of E in G , s h o w i n g , b y Lemma 2.4-1, that G has t h e s u b n o r m a l i n t e r s e c t i o n p r o p e r t y .
2 . 4 4 E X A M P L E ( [ 2 4 ] , L e m m a 2 ( i i i ) ) . C o n s i d e r t h e s t a n d a r d
r e s t r i c t e d w r e a t h p r o d u c t G = X vnc I o f a p - c y c l e X and a
q u a s i - c y c l i c p - g r o u p Y , w h e r e p is a n y p r i m e . D e n o t e b y B t h e
b a s e g r o u p o f t h i s w r e a t h p r o d u c t . L e t T b e t h e r i n g o f e n d o m o r p h i s m s
o f B ; t h e n f o r a n y t € T , ( t - 1 ) ^ = - 1 , s i n c e B h a s
e x p o n e n t p . If h i B a n d t i T ve d e n o t e b y [b, t] t h e
e l e m e n t = a n d b y [ B , t] t h e s u b g r o u p o f B .
W e m a y r e g a r d J a s a m u l t i p l i c a t i v e l y c l o s e d s u b s e t of T . If
v
y ^ y t h e n f o r a n y p o s i t i v e i n t e g e r r , y - 1 c a n b e f a c t o r i s e d
a s Piy){y-1) f o r s o m e p o l y n o m i a l Piy) . S i n c e y h a s f i n i t e o r d e r ,
it is e a s i l y s e e n t h a t t h e s u b g r o u p [ B, ( y) ] of G is just [ B, y] .
F o r e a c h p o s i t i v e i n t e g e r m , c h o o s e an e l e m e n t y^ in Y of
o r d e r p ^ . T h e n if Y ^ d e n o t e s ^ ^ ^ ^ J w e h a v e
m [y -l)^ yBYP = B ^
' m
/ - I
- b" -1 ,
b u t
m-1
m - l y ^ - 1
yBYP - B ^ m
T h i s m e a n s t h a t , s i n c e G - YB a n d Y is a b e l i a n .
but
m-1
yGY^ n B 1 . m
It follows that each Y^ is subnormal in G , and that
p"-^
< S(G :g S p"
.Thus there is certainly no bound for the defects of the subnormal subgroups of G .
Now we show that, for any subgroup H of G , the standard
series of H in G becomes stationary after finitely many steps. If HB < G , then HB - Y^B for some positive integer m , or else
HB = B . If HB = Y^B , then HB is the join of a normal abelian
subgroup B and a subnormal abelian subgroup Y^ , so it is nilpotent
by Lemma 2.22. H is then subnormal in HB , which is normal in G , so the standard series of 5 in G reaches H after finitely many steps. If HB - B the same is true. The only outstanding possibility is HB = G .
If this is the case, consider [B, Y] . This subgroup is
generated by elements of the form b^'^ ^^ with b i B , y i. Y . Now
-y = -y^ for some y^ ^ Y , thus
=b = b ^ € yBY^
[B, Y] = [B, I, I]
Now
[B, m = [B, Bff] since B is abelian = [B, BY] since BH = G = BY = [B, J] since B is al?elian.
Similarly [ B, H, H] = [ B, Y, Y] , so that [ B, H, H] = [ B, H\ .
Hence
^ H[G, H]
- H[B, H]
= a B, H, H] = H^'^ .
Thus even in the case HB - G , the standard series of ff in G is well-behaved. Hence G has the subnormal intersection property, by Lemma 2.1+1.
A substantial part of this thesis is taken up with the question: for what groups does the possession of the subnormal intersection property imply the existence of a bound on the subnormal indices? Example 2.44 shows that this implication does not hold even for metabelian p-groups.
w e can p r o v e t h e f o l l o w i n g s t r o n g e r r e s u l t .
2.45 L E M M A . If G has a normal subgroup N which satisfies the minimal condition on subnormal subgroups, and G/N has the
subnormal intersection property, then G has this property also.
P r o o f : Let H be any subgroup of G . By Lemma 2.41 and r e m a r k 1.23 ( c ) , t h e r e is an integer k > 0 such that
ff'^'^il/ = , since G/N has the s u b n o r m a l intersection
p r o p e r t y . But since N satisfies the m i n i m a l condition on subnormal s u b g r o u p s , t h e r e is an integer m > k such that
^ G , m ^ ^ ^ ^ ^ ^ ^ G ^ is also t r u e , b y a
f a m i l i a r a r g u m e n t w e h a v e j p - j p ^ This proves the r e s u l t , by Lemma 2 . 4 1 .
R E M A R K . In [25] (Lemma 3 . 2 ) Robinson proves a stronger result t h a n that m e n t i o n e d in t h e context of Lemma 2 . 4 5 , n a m e l y that a group w h i c h s a t i s f i e s the m i n i m a l c o n d i t i o n on s u b n o r m a l subgroups even has a b o u n d for its s u b n o r m a l i n d i c e s . This somewhat surprising result w i l l be f u r t h e r s t r e n g t h e n e d in 4 . 4 3 , w h e r e w e w i l l h a v e enough m a c h i n e r y a v a i l a b l e to prove the obvious a n a l o g u e of Lemma 2 . 4 5 .
2.5 Joins of Subnormal Subgroups
W e c o n c l u d e this c h a p t e r w i t h a discussion of the theory of joins of s u b n o r m a l s u b g r o u p s , as p r o p o u n d e d b y Robinson in [ 23] . The
t r e a t m e n t w i l l be s k e t c h y , as this topic occupies only a p e r i p h e r a l p o s i t i o n in t h e t h e s i s .
that in any group the join of two subnormal subgroups is a subnormal subgroup. In Theorems 6.1 and 6.2 of [23] Robinson constructs a soluble group of derived length 3 and a finitely-generated soluble group of derived length 4 in which this fails to happen.
However, the class of groups in which the join of two subnormal subgroups is always a subnormal subgroup is too large to be of much concern to us here. Indeed most of the groups with which we deal lie in the much narrower class of groups in which the join of an arbil^rary family of subnormal subgroups is a subnormal subgroup. We will say that the groups in the latter class have the subnormal join property.
Example 2.44- provides us with a group which does not have the subnormal join property, for although (in the notation of 2.44)
each of the subgroups Y^ is subnormal, their join Y = ^^^^ : m > l}
cannot be a subnormal subgroup, for if it were G would be nilpotent by Lemma 2.22. This group, therefore, has the subnormal intersection property but not the subnormal join property. We will see from 2.51 below that any group satisfying the maximal condition on subgroups has the subnormal join property. Thus the infinite dihedral group (2.35) has this property but fails to have the subnormal intersection property.
If a group G has the property that the join of any two subnorma^l subgroup is a subnormal subgroup, we can consider the set of all
subnormal subgroups of G as a lattice under the operations of
s u b g r o u p s of G . G r o u p s w i t h t h e s e p r o p e r t i e s need not h a v e a b o u n d on t h e i r s u b n o r m a l i n d i c e s : for i n s t a n c e , t h e H e i n e k e n - M o h a m e d
e x a m p l e ([ 14] ) m e n t i o n e d b e f o r e 2.36 has b o t h t h e s e p r o p e r t i e s , but by-T h e o r e m 2.36 it c a n n o t h a v e a b o u n d for t h e d e f e c t s of its s u b n o r m a l s u b g r o u p s , s i n c e it is not n i l p o t e n t . H o w e v e r , g r o u p s w i t h t h e s e two p r o p e r t i e s do b e h a v e s o m e w h a t s i m i l a r l y to g r o u p s w i t h b o u n d e d s u b n o r m a l i n d i c e s , as e v i d e n c e d , for e x a m p l e , in t h e a t t a c h e d paper [A] .
We s t a t e , w i t h o u t p r o o f , a b a s i c l e m m a .
2.51 L E M M A ([2 3] , Lemma 8 . 1 ) . A group G has the subnormal join -proiperty if and only if the union of any asoending chain of subnoimal subgroups of G is a subnormal subgroup of G .
S i n c e o u r n e x t l e m m a , t h o u g h w e l l - k n o w n , is not p r o v e d in the l i t e r a t u r e , w e i n c l u d e a p r o o f .
2.52 L E M M A . A group with a bound on its subnormal indices has
the subnormal ;ioin property.
P r o o f : Let (J b e a g r o u p w i t h a b o u n d r on its s u b n o r m a l i n d i c e s . Let [H . : i ^ I) b e an a s c e n d i n g chain of s u b n o r m a l
t-s u b g r o u p t-s of G , and let H be t h e i r u n i o n . We p r o v e b y induction
t h a t for e a c h n o n - n e g a t i v e i n t e g e r n ,
yGlP S : i e l | .
T h i s is t r i v i a l l y t r u e for n = 0 . If n > 0 and t h e statement is
h] g i yGlP ^, h i H) ,
so t h a t it w i l l s u f f i c e to show that each element of the form
[g.h] : g i yGH^'^, h ^ H ,
l i e s in some yGlf] . But b y the induction h y p o t h e s i s , g ^ yGlf.'^
^ 3 for some Q ^ I \ also h ^ E-^ for some k I . There is an
e l e m e n t m ^ I such t h a t (h, H.) S H , for either H. H, or else
J m o k
H. S H . . It is then c l e a r that [ g , h] € yGli^ as r e q u i r e d .
^ CI 171
Now it f o l l o w s that
yGlf S : i € l | < : i € J } ,
s i n c e each H. is s u b n o r m a l of defect at m o s t r . That i s , yGlf S H ,
so H is s u b n o r m a l in G , By 2 . 5 1 , G has the s u b n o r m a l join p r o p e r t y .
F i n a l l y . w e p r o v e a lemma w h i c h w i l l be of u s e in later c h a p t e r s .
2.53 LEMMA ([2 3 ] , Lemma 2 . 2 ) . Let H and K be subnormal
subgroups pf a group G . If H^ ^ H , then J = i H , K) = HK is subnormal in G and s(G : J) 5 s{G : H)s(G : K) .
P r o o f : Let r and s be t h e d e f e c t s of H and K r e s p e c t i v e l y .
D e n o t e by H . the subgroup H ^ . Then if ZfJ = H . , it follows 1 L U
H.K KH. H.
[H. y = H'' -- H -- E^ = E. , ,
using E^ - E . Since E - G it is clear that each term E. in the U
standard series of E hi G is normalised by K . Thus for each i with 0 ^ i < r , is normal in E .K , and so by remark 2.21
"Z- +1
(c), E. K has defect at most s in E .K . A simple summation then
t t -T O R S I O N - F R E E N E S S A N D T T- R A D I C A B I L I T Y
In this chapter we introduce and develop several useful concepts
which enable us to prove a sequence of lemmas needed in later w o r k .
The first section is devoted to a general discussion of series in a
g r o u p , with specific reference to Zi4-groups. In Section 3.2 we
discuss TT-torsion-freeness in some restricted situations. Sections
3.3 and 3.1+ deal with quasi-7T-radicable and TT-radicable groups, and
their place in the theory of subnormal subgroups. Finally, in Section
3 . 5 , we prove the promised succession of lemmas. These results,
mainly concerned with automorphisms of groups with the properties
discussed in 3.2 and 3 . 3 , will be of fundamental importance in
Chapters 5 and 6 . In the early parts of the chapter many of the proofs
are omitted for brevity.
3.1 Series and central series
A basic concept of infinite group theory is that of a series or
n o r m a l system in a group. (See, for example, [/7], p . 171). In this
section we present a treatment which follows that of Hall [11].
3 . 1 1 D E F I N I T I O N . Let G be a group and a linearly ordered
set. By a series in G of type Q, we mean a set
{ A ^ , v ^ : a e f^}
of pairs of subgroups of G , with the following properties:
(i) \J ^ is a normal subgroup of A ^ for each a in ;
(ii) whenever a < T ;
(iii) ff - 1 : ° ^ •
H e r e , if Z and 7 are subsets of G , X - Y denotes the
Another easy consequence of the conditions is (iv) for each O ^ fl ,
(a) V a = : T < o} ; (b) A ^ = n { V ^ : T > a} .
The series (>'0 is said to be invariant when, for each a in fi , A ^ is a normal subgroup of G , or, equivalently, when, for each a
in fl , \/ g is a normal subgroup of G .
The series (*) is said to be central when, for each a in , L'^a. € •
Clearly any central series is invariant.
The more familiar concepts of ascending and descending series can be obtained as special cases of this general type of series.
(A) If ^ is well-ordered then (''0 becomes an ascending series. In this case we may take 9, to be the set of all ordinals Q < p for the ordinal p which is the order-type of fi . Then we have
A = V , for a t 1 < p , and if we define \/ = G the A
a ^ a+1 ^ p o become superfluous, and the series takes the form
{ V a = ° - '
with = 1 , V(j normal in N/a+l fo^ < P > V p = ^ . ^^d, for every limit ordinal y S p ,
V ^ = : a < y} .
(B) If is inversely well-ordered, then (") becomes a descending series. Again we may take 9 to be the set of all
- V ^ whenever x < a .
Then we have V a ^ a+1 + 1 < P , and if we define
^ p = 1 "the y ^ become superfluous, and the series takes the
form
{ A ^ : a 5 p} ,
with A ^ = G , A normal in A ^ for each a < p , A ^ = l
and, for every limit ordinal y 5 p ,
A ^ = n { A ^ : a < y} .
The notion of series will be used strongly in Section 5.4, but for this chapter we require it in the following well-known definition.
(See [17], p. 218.)
3.12 DEFINITION. A ZA-group is a group which has an ascending central series.
We make two remarks on this definition.
(a) Any subgroup or quotient group of a Zi4-group is a Zi4-group. (b) The centre of a Zi4-group is non-trivial.
In any group G we can define an ascending chain of subgroups of G , its upper oentral series , by taking
S^Cff) - 1 ,
^(G) to be the centre of for each positive,
non-limit ordinal Oi , and
^^((J) = ^{^^((J) : a < y}
for each limit ordinal y . There will of course be a first ordinal P , depending on G , such that ^^(G) = • T^® upper central
series of G is then
only if, for this ordinal 3 , ^^(.G) = G , that is, the upper central
series of G is an ascending series in the strict sense of 3.11 (A).
(G is of course nilpotent if and only if this ordinal 3 is finite.)
In later stages of the chapter it will be convenient to have
available some simple results on Z/4-groups. The first of these is
well known; for the (easy) proof we refer the reader to [29], (1.51).
3.13 LEMMA. Let N be a non-trivial normal subgroup of a
ZA-group G . Then N n ^^(G) is non-trivial. (Eere is^ as
above, the centre of G .)
Our next lemma is a well-known result of Mal'cev. For the (not
so easy) proof see [/7], p. 223.
3.14 LEMMA ([7S]). A finitely-generated ZA-group is nilpotent.
Before stating the last result of this section, it will be necessary
to introduce some standard terminology which will be used throughout
this and subsequent chapters.
We will denote by TT a non-empty set of prime natural numbers,
and by it' its complement in the set of all primes. A positive
integer k will be called a "n-number if each prime divisor of k
lies in fT : we may think of the set of TT-numbers as the multiplicative
sub-semigroup of the positive integers generated by the set TT ^ {l} .
In a group, an element will be termed a Tl-element if its order is a
TT-number. A group in which every element is a TT-element will be
called a "n-group; if no element, other than 1 , is a TT-element the
group will be said to be T^-torsion-free. If tt consists of only one
prime, p say, we will use such terms as p-group, p-torsion-free.
Indeed a group is TT-torsion-free if and only if it is p-torsion-free
e l e m e n t a r y p r o p e r t i e s of n i l p o t e n t groups ( s e e , for e x a m p l e . T h e o r e m 1.10 of [ 9 ] ) .
3.15 L E M M A . Let G be a ZA-group and rr a non-empty set of primes. Then the set of ^-elements of G is a (fully invariant)
subgroup of G .
In any g r o u p , t h i s s u b g r o u p , if it e x i s t s , is called the iT-torsion s u b g r o u p .
3.2 TT^torsion-freeness
In t h i s short s e c t i o n w e r e c o r d , for easy r e f e r e n c e , some b a s i c r e s u l t s on 7T-torsion-free g r o u p s . The first of t h e s e is due to M a l ' c e v ; a p r o o f can b e found in [ 2 9 ] , ( 1 . 6 3 ) .
3.21 L E M M A . Let G be a group whose centre is
torsion-free for some non-empty set of primes tt . Then for any ordinal a C (G) is i\-torsion-free.
Cx+_L Ct
A s i m p l e t r a n s f i n i t e - i n d u c t i o n leads to the following c o r o l l a r y . 3.22 C O R O L L A R Y . Let G be a group with upper central series
: a 5 3 } .
If is i\-torsion-free for some non-empty set of primes IT ^
then for any ordinals X , y with y < X 5 3 ^^
torsion-free. In particular, if G is a ZA-group, G itself is
•n-torsion-free.
A l t h o u g h w e w i l l b e m a i n l y c o n c e r n e d w i t h n i l p o t e n t (often a b e l i a n ) 7 T- t o r s i o n - f r e e g r o u p s , w e r e m a i n in the m o r e g e n e r a l
primes. Then G is n-tovsion-free if and only if, for each p in
TT ccP = y^ with X, y in G implies x = y .
N o t e that t h e c o n d i t i o n could b e r e p l a c e d b y the r e q u i r e m e n t that k k
f o r each iT-number k , x = y w i t h x, y in G implies x - y .
3.3 Quasi-TT-radicabil ity
B e f o r e m a k i n g t h e first d e f i n i t i o n of this s e c t i o n , w e need to i n t r o d u c e some m o r e n o t a t i o n . If G is a group and k a p o s i t i v e k k i n t e g e r , w e w i l l d e n o t e b y G the subgroup ig : g ^ G) of G .
k
For each k , G is a fully invariant s u b g r o u p of G .
We are n o w in a p o s i t i o n to d e f i n e and c o m p a r e two important s t a n d a r d c o n c e p t s .
3.31 D E F I N I T I O N . If IT is a n o n - e m p t y set of p r i m e s , a group G is said to b e quasi-T^-radioable i f , for each iT-number k ,
G - G , that i s , each element of G can b e expressed as a product of fe-th p o w e r s .
3.32 D E F I N I T I O N . If TT is a n o n - e m p t y set of p r i m e s , a group G is said to b e i\-Tadioable i f , for each TT-number k , each element of G is t h e k-th p o w e r o f some element of G .
We r e m a r k that w h e n TT is the set of all primes w e w i l l use the t e r m s " q u a s i - r a d i c a b l e " ("Cernikov c o m p l e t e " ) and "radicable" ("complete' o r , for a b e l i a n g r o u p s , " d i v i s i b l e " ) .
It is c l e a r that D e f i n i t i o n 3.32 loses none of its force if w e r e p l a c e "for each TT-number k " b y "for each prime p in TT " . In c o n t r a s t to t h i s . D e f i n i t i o n 3.31 cannot b e w e a k e n e d in this way
p r i m e p in TT = { 2 , 3 , 5} , b e c a u s e for each r > 0 , F^ is a l w a y s a n o n - t r i v i a l n o r m a l subgroup of F . But the fact that
30
F = 1 shows that F is not quasi-ir-radicable, The following simple l e m m a shows t h a t if w e r e s t r i c t our attention to soluble groups this d i f f i c u l t y d i s a p p e a r s .
3.33 L E M M A . A soluble group G is quasi-'^i-vadiaable, for some
non-empty set of primes it if and only if G - G^ for each p in TT .
P r o o f : Only the s u f f i c i e n c y of the condition is in q u e s t i o n .
Suppose G = Cp for each p in '~7T , and let k b e any TT-number. k
If w e w r i t e H - G/G t h e n H is a soluble group of exponent
d i v i d i n g k and H
-hP for each p in TT . T h e n H/H' is an a b e l i a n g r o u p w i t h the same p r o p e r t i e s as H , so H/H' is t r i v i a l . H e n c e H is t r i v i a l and G - G , proving the l e m m a .
For s o l u b l e groups the criterion of 3.33 w i l l b e used (often i m p l i c i t l y ) as a s u b s t i t u t e for D e f i n i t i o n 3 . 3 1 .
We now state a t r i v i a l , b u t u s e f u l , c h a r a c t e r i s a t i o n of
quasi-TT-radicability. (Both 3.34 and 3.33 are p r e s u m a b l y w e l l - k n o w n . ) 3.34 L E M M A . A group G is quasi-'^-radiaable, for some
non-empty set of primes tt if and only if G has no proper normal subgroup N suah that the exponent of G/N is a "^-number.
It is clear from a comparison of Definitions 3.31 and 3.32 that 7T-radicability implies quasi-TT-radicability. Indeed it is easy to see t h a t a g r o u p G is TT-radicable if and o n l y if it is quasi-n-radicable
3 . 3 5 E X A M P L E ( [ 2 9 ] , ( 6 . 4 ) ) . L e t G b e t h e s t a n d a r d r e s t r i c t e d
w r e a t h p r o d u c t o f t w o q u a s i - c y c l i c p - g r o u p s , f o r a g i v e n p r i m e p .
C l e a r l y G h a s n o p r o p e r q u o t i e n t s o f f i n i t e e x p o n e n t a n d s o , b y
3.3M-, is q u a s i - r a d i c a b l e • H o w e v e r it is r o u t i n e t o s h o w (see [ 2 9 ] )
t h a t G is n o t e v e n p - r a d i c a b l e .
3 . 3 6 E X A M P L E . If p is a n o d d prime^, d e f i n e a g r o u p G^ b y
G^ = ia, h : aP = b^ = (ab)^ = 1 ) .
S i n c e b = b^ a n d ab = iab)^ , G^ is q u a s i - p - r a d i c a b l e . But a
c a n h a v e n o p - t h r o o t s i n c e | | = 2p , so G ^ is n o t p - r a d i c a b l e .
T h e p r o o f o f o u r n e x t l e m m a i n v o l v e s a m u c h - u s e d t e c h n i q u e .
3 . 3 7 L E M M A . Let TI be a non-empty set of primes- for which IT'
is also non-empty. Then if G is a tj'-group^ G is l\-radioable;
and conversely, if G is a •n-radicable - group of finite exponent k ,
k must be a "n'-number.
P r o o f : L e t m b e a TT-number. T h e n f o r a n y g in G , t h e
o r d e r r o f ^ is c o p r i m e t o m , as G is a TT'-group. T h u s t h e r e
a r e i n t e g e r s s a n d t s u c h t h a t sr + tm = 1 . T h e n
sr+tm ( t\m g = g = J »
t h a t i s , g is a n m - t h p o w e r . T h i s p r o v e s t h a t G is iT-radicable.
C o n v e r s e l y , if G is TT-radicable a n d s o m e p r i m e p in it
d i v i d e s t h e e x p o n e n t k of G , t h e n fe = p n f o r s o m e i n t e g e r n
w i t h 0 < n < k . N o w f o r e a c h ^ in G t h e r e is a n e l e m e n t h of
G w i t h g = h^ . T h e n
/ = /zP'^ = /z^ = 1 ,
T h e f o l l o w i n g i m p o r t a n t t h e o r e m , a g e n e r a l i s a t i o n of a r e s u l t of C e r n i k o v , shows that for Z;4-groups t h e situations arising in examples 3.35 and 3.36 c a n n o t o c c u r . We w i l l o m i t t h e p r o o f for b r e v i t y , since it is an easy g e n e r a l i s a t i o n of that given in [ 2 9 ] , Theorem (6.41) (see also [ 7 7 ] , p p . 2 3 4 , 2 3 8 ) .
3.38 T H E O R E M . Let G be a quasi-i\-radiaable ZA-gvoup, for some non-empty set of primes TT . Then
(i) G is T\-radioahle;
Hi) the Ti-torsion subgroup of G is i\-radioable and lies in the centre of G ;
(Hi) if ^^d are any two terms of the upper
central series of G , with a > g then is
i\-radioable.
T h e r e is a v e r y u s e f u l c o r o l l a r y .
3.39 C O R O L L A R Y . A quasi-Ti-radicdble ZA-group which is a "n-group is abelian.
3.4 TT-radicabil ity
In t h i s s e c t i o n w e c o n t i n u e our investigations of quasi-7T-radicability and TT-radicability.
In a n y g r o u p G , t h e join of any family of quasi-TT-radicable s u b g r o u p s is a g a i n a quasi-TT-radicable s u b g r o u p , for obvious r e a s o n s . In c o n t r a s t , a s i m i l a r statement for TT-radicable subgroups does not h o l d , for if p is an odd p r i m e t h e group G ^ of Example 3.36 is the
join of two p - r a d i c a b l e s u b g r o u p s <b) and ( ab ) , yet G^ is not
primes 7T , every group G has a unique maximal quasi-TT-radicable
subgroup , which is clearly a fully invariant subgroup of G ,
since each of its quotients remains quasi-7T-radicable. Moreover the
trivial fact that an extension of a quasi-U-radicable group by another
is a quasi-TT-radicable group forces is to conclude that G/Qirr) has
no non-trivial quasi-lT-radicable subgroups. A group with the latter
property, namely that its maximal quasi-iT-radicable subgroup is
trivial, is termed TT-reduced. If tt is the set of all primes, we
use the term reduced.
Although we have seen that in general a group need not have a
unique maximal TT-radicable subgroup. Theorem 3.38 ensures that in the
case of a Z^-group such a subgroup will exist.
The first lemma of this section records, for reference, some
simple connections between these ideas and those of 3.2.
3.41 LEMMA. If is a non-empty set of primes^ and Qi'^) is the maximal quasi-'^-radioable subgroup of a group G „ then G/Q('n)
is -torsion-free. If, in addition, G is a -n-torsion-free
ZA-group, G/Qi"^) is in fact torsion-free.
Proof: Since any IT'-group is iT-radicable by 3.37, and G/Q{t:)
is TT-reduced, the first part is immediate. To prove the second, we
need only show that G / Q W is Tf-torsion-free. Suppose, therefore,
that X (i G with x^ ^ SC^r) for some TT-number k . Now ^(tt) is
TT-radicable by Theorem 3.38, so there is an element y in QiT^) such
that x^ = y^ . By Lemma 3.23 we have x = y , that is x € Q(tt) . This shows that G/Qi^) is TT-torsion-free, as required.
It is convenient in this context to include a useful result due
