• Aucun résultat trouvé

Chapter VIII. Chapter VII. Chapter VI. Chapter V. III. II. Chapter Chapter Chapter Chapter New York University, New York, U.S.A. SOME ASPECTS OF GROUPS WITH UNIQUE ROOTS

N/A
N/A
Protected

Academic year: 2022

Partager "Chapter VIII. Chapter VII. Chapter VI. Chapter V. III. II. Chapter Chapter Chapter Chapter New York University, New York, U.S.A. SOME ASPECTS OF GROUPS WITH UNIQUE ROOTS"

Copied!
87
0
0

Texte intégral

(1)

SOME ASPECTS OF GROUPS WITH UNIQUE ROOTS

BY

G I L B E R T B A U M S L A G New York University, New York, U.S.A.

Contents

Page

I n t r o d u c t i o n . . . 217

A c k n o w l e d g e m e n t s . . . 220

N o t a t i o n . . . 220

P r e l i m i n a r i e s . . . 221

PART I. Chapter Chapter Chapter Chapter I . D e f i n i t i o n s a n d g e n e r a l i t i e s . . . 222

I I . L o c a l l y n i l p o t e n t g r o u p s Z A - g r o u p s a n d n i l p o t e n t g r o u p s . . . 229

I I I . C o n s t r u c t i o n of U ~ - g r o u p s , E ~ g r o u p s a n d D ~ - g r o u p s . . . 237

V I . E x t e n s i o n s of U ~ g r o u p s , E ~ - g r o u p s a n d D ~ - g r o u p s . . . 246

PART II. Chapter V. T h e a b s t r a c t D ~ - f r e e g r o u p . . . 252

Chapter VI. T h e f u n d a m e n t a l e m b e d d i n g t h e o r e m . . . 256

Chapter VII. E m b e d d i n g of U ~ - g r o u p s i n D o - g r o u p s . . . 267

Chapter V I I I . D ~ - f r e e g r o u p s a n d D ~ f r e e p r o d u c t s of D ~ - g r o u p s . . . 279

Introduction

1. T h e m u l t i p l i c a t i v e n o t a t i o n e m p l o y e d i n t h e s t u d y of g r o u p s l e a d s , i n a n a t u r a l w a y t o t h e n o t i o n of r o o t s i n g r o u p s . T h u s if n is a p o s i t i v e i n t e g e r a n d g is a n e l e m e n t of a g r o u p G t h e n a s o l u t i o n x of t h e e q u a t i o n

X n ~ g

(2)

2 ] 8 GILBERT BAUMSLAG

is called an n-th root of g. I n general g m a y not h a v e an nth root; on the other hand it m a y have more than one. If every element in G has an nth root for every positive integer n then G is called a divisible or complete group.

Divisible groups appeared first in the theory of abelian groups; one of the classical theorems in this connection asserts that every abelian group can be embedded in a divisible group, which is also abelian. I n recent years a large number of Russian mathematicians have carried out investigations of particular classes of divisible groups which have certain commutativity properties. Thus ~ernikov [7] has studied divisible groups with an ascending central series which sweeps out the whole group, and Mal'cev [25], [26] has studied locally nilpotent groups which are divisible. I n particular Marcev [25] proved the beautiful theorem t h a t every torsion-free locally nilpotent group can be embedded in a torsion-flee locally nilpotent divisible group. Mal'cev [25] proved also that the extraction of roots is unique in a torsion-free locally nilpotent group G; in other words for any x, y E G and a n y non-zero integer n, the equation

x n __ yn

implies x = y. Groups with this property were given the name of R-groups b y Kontorovi6 [18], [19] who extended some earlier work b y Baer [1] on torsion-free abelian groups to the larger class of R-groups.

We shall be concerned here with three kinds of groups which contain divisible groups, R-groups and divisible R-groups as special cases: For each non-empty set of primes ~o we define 3 associated classes of groups. Thus E~ denotes the class of groups in which pth roots exist for all pEco, and U~ denotes the class of groups in which p t h roots are unique for all p Co); consequently E~ N U~ is the class of those groups in which pth roots not only exist, but are unique--we shall henceforth denote the class E~ n U~ by D~. If G E Eo~ we call G an E~-group; on the other hand, if G E U ~ we call G a U~-group; if G E D ~ we call G a D~-group. So in the particular case where ~o coincides with the set of all primes, an E~-group is a divisible group, a U~-group is an R-group and a D~-group is a divisible R-group.

2. Part I of this work is concerned with miscellaneous properties of E~-groups, Uo~- groups and D~-groups. Here (and throughout this paper) we are motivated by the concepts of universal algebra to fix some of our attention on certain subgroups of E~-groups and D~-groups. One of our results in this connection is t h a t the derived group of a locally nil- potent D~-group is itself a D~-group. We concern ourselves also with various "extension"

problems. Thus we prove t h a t an extension of a ZA-group in E~ b y a periodic group in U~ belongs to E~. A similar, although unrelated, result is the following: A locally nilpotent

(3)

SOME ASPECTS OF G R O U P S W I T H U N I Q U E I~OOTS 2 1 9

group which is an extension of a D,-group b y a D~-group is a Do-group. Another topic treated in Part I is the construction of U~-groups, Eo-groups and D~-groups.

3. The class of groups Do forms a "variety" of algebras in the sense of P. Hall; equi- valently, it is equationally definable. To see this we introduce a set f~o of unitary operators in a fixed one-to-one correspondence with co, ;r Cf2o corresponding to p Eco; these, together with the group operations, are to be the operators of the variety Do. The laws of Do are the group laws together with the further laws, 2 for each p Co:

( x ~ F - x , x ' ~ = x .

I t is easy to see t h a t a group which admits the operators Jr (in the sense of Higgins [14]) is a D~-group and conversely.

Now it follows from general results on varieties of algebras (Birkhoff [5]) t h a t there are free algebras in the variety Do; we call these free algebras D~-free groups. Although the existence of D~-free groups is thus taken care of there are still a large number of im- portant questions that the bare knowledge of the existence will not answer.

The notion of a free group is of vital importance in the theory of groups. I t seems likely t h a t a Do-free group will play as important a role in the study of D~-groups. A simple

"normM form" with reference to a fixed set of free generators is available for the elements of a free group. This normal form facilitates the proof of a large number of theorems about free groups. One of the difficulties involved in the study of Do-free groups is the absence, at first sight at least, of a useful simple normal form. This makes difficult the proof of such a seemingly obvious, and in fact true, assertion as: A Do-free group is torsion-free. The most important part of this paper seems to be P a r t I I in which a Do-free group is con- structed as the union of an ascending sequence of U~-groups; these Uo-groups are them- selves generalised free products of Uo-groups with a single amalgamation. The complica- tions involved in the study of generMised free products therefore occur here and so this construction serves also to illustrate the difficulties inherent in such groups. However we are able to utilise it to investigate the structure of D~-free groups. I n particular we prove the surprising result t h a t every Eo-group is the homomorphic image of a Do-free group.

4. B. H. Neumann [27] has shown that a free product of groups can be defined by means of a homomorphism property. We define, analogously, a Do-free product of

Do~-

groups. The existence of this product is established b y an actual construction which enables us to derive some of the properties of Do-free products.

5. Only a small number of the questions t h a t present themselves in connection with Do-free groups and Do-free products of Do-groups have been answered here. The solution

(4)

2 2 0 GILBERT BAUMSLAG

of some of these questions seem, b y the very nature of the groups involved, to offer more t h a n token resistance. H o w e v e r the methods developed here m a k e possible, at least theo- retically, the solution of most of these questions; we hope to deal with further questions in later papers.

6. Acknowledgments.

I t a k e this opportunity to express m y gratitude and appreciation to m y Parents, without whose help and encouragement this o p p o r t u n i t y for further study would have been b o t h unwanted and impossible.

I t is a v e r y great pleasure to acknowledge the help of Dr. B. H. N e u m a n n who, with his ever-ready advice, criticism and creative remarks, has made it a privilege and a delight to have him as a supervisor.

7. Notation.

F o r the reader's convenience we list some of the notations used.

( D

gp(X, R) [g, h]

[G, H]

a' = r (G) = Pl (G) P~+I(G)

gh

g h l + h ~ + 9 9 + h n g m U

g~

~(G)

IGI ISI

c (s, G) C(S) C (8, G) C(s) N (z, G) N(z) n m (S)

a n o n - e m p t y set of primes.

the group generated b y the set X with defining relations R.

the c o m m u t a t o r

g-lh-lgh

of g and h.

the group generated b y the commutators [g, h], g E G, h E H.

[G, G], the derived group of G.

[F~ (G), Ft (G)], the i + 1st derived group of G.

the transform

h-lgh

of g b y h, g,

hGG.

gh,.gh,.., ghn

(g, hi ' h2 . . . hn in the group G).

gh. gh ... ga, with m > 0 an integer.

m

the image of g under the homomorphism ~ of the group G.

g~. gV, where here T and ~p are homomorphisms of the group G into the group H.

the centre of the group G.

the order of the group G.

the cardinality of the set S.

the centraliser of the subset S in the group G.

the centraliser of the subset S in the group G, where G here is understood.

c ({8}, G).

c({8}).

the normaliser of the subset S in the group G.

the normaliser of the subset S in the group G, where here G is understood.

the normal closure of the subset S in the group G, i.e. the intersection of all normal subgroups of G containing S.

(5)

H * A ~

).cA

F l-x- _F~ ~ ... * F n

{A.B;

1-I A~

2cA

A •

{14 • 8; H}

H <~G H < G H<a G

z(p ~)

(m~ n )

S - T

SOME ASPECTS OF GROUPS ~VITH UNIQUE ROOTS 2 2 1

the free product of the groups A~ (2 EA).

the free product of F1, F 2 .. . . . Fn.

the free product of A and B with amalgamated subgroup H.

the (restricted) direct product of the A~ (~ EA).

the direct product of A and B.

the direct product of A and B with amalgamated subgroup H.

H is a subgroup of G.

H is a proper subgroup of G (here we do not exclude the possibility H = I ) .

H is a normal subgroup of G.

the multiplicative group of all p%h roots of unity, where p is a fixed prime and n ranges over the non-negative integers.

the greatest common divisor of the integers m and n.

the set-theoretical difference between S and T, where T is a subset of S.

8. Preliminaries.

Suppose t h a t / , g and h are elements of a group G. Then the following relations between commutators hold:

[g/, h] - [g, hit[l, h] and [g,/hi = [g, h] [g, /]h.

We refer the reader to Kurosh [21] vol. 2 for the definitions of a free group, a free product, upper central series of a group, lower central series of a group, nilpotent group, ZA-group, locally nilpotent group, soluble group and derived series of a group.

We say t h a t G is an extension of A b y B if A <~ G and G / A ~ B; an extension is called central if A <~ ~ (G).

(6)

222 G I L B E R T B A U M S L A G

P A R T I

CHAPTER I Definitions and generalities

9. L e t H be a s u b g r o u p of a n a r b i t r a r y group G. T h e n we call H a n co-subgroup(1) of G if t h e r e l a t i o n g~EH implies gEH for a n y pair g a n d p, w i t h gEG a n d p E w . I f H<~ G we call H a n w-ideal(e) of G if G/H E Uo). These t w o concepts are relative; however, if t h e r e is n o a m b i g u i t y i n v o l v e d we shall s i m p l y call a n w-subgroup o/G a n ~o-subgroup a n d a n co-

ideal o] G a n o~-ideal.

LEMlVIA 9.1 The intersection o/w-subgroups is an w-subgroup.

Proo/. L e t {H~} be a set of ~o-subgroups of G, ~ r a n g i n g over a n i n d e x set A. If g i n

G, p i n w are such t h a t g~E N H~, t h e n gPE H~ for all g E A, so g E H~ for all ~ E A. T h u s

acEA

r l H~ is a n m-subgroup.

cr

LEMMA 9.2. The intersection o/m-ideals is an ~o-ideal.

Proo/. L e t {H~) be a set of w-ideals of G, ~ r a n g i n g over a n i n d e x set A, a n d let H b e t h e i r intersection. T h e n H <~ G. F u r t h e r m o r e G/H E Uo,. F o r suppose p E o~, gl, gz E G a n d (glH) p = (g2H) p. T h e n (glH~) p = (g2H~) p a n d hence glH~ =g2H~ for e v e r y a E A . I n o t h e r words gig2-1 E H~ for e v e r y a E A a n d hence gig21 E H. C o n s e q u e n t l y gl H = g2H.

Suppose t h a t H is a n o r m a l w - s u b g r o u p of G. I t does n o t follow, i n general, t h a t H is a n (o-ideal of G, e v e n if we presuppose t h a t G E U~. F o r let

C = g p ( a , b; a 2 =b2);

i t c a n be verified d i r e c t l y t h a t C c o n t a i n s n o e l e m e n t s of order 2 (and i n fact n o e l e m e n t s of finite order). N e x t p r e s e n t C as a factor group of a free group G b y a n o r m a l s u b g r o u p H :

C~= G/H.

N o w a free g r o u p is a U(2~-group (it is i n fact a n R-group, see e.g. ] ( o n t o r o v i 5 [18] or T h e o r e m 17.2 of this paper). I t is clear t h a t H is a ( 2 ) - s u b g r o u p of G because C c o n t a i n s n o e l e m e n t s of order 2. So here we h a v e a n o r m a l o)-subgroup of a U~-group which is n o t a n (1) I n the case where eo coincides with the set of all primes and GC U~ an o)-subgroup of G is called isolated by Kurosh [21] vol. 2, p. 243.

(2) In the case where GEDo) the co-ideals defined here coincide with the ideals defined by Hig- gins [14].

(7)

SOME ASeECTS OF OROVPS WIT~ U~IQUE ROOTS 223 w-ideal. For D~-groups it is more difficult to make an example of such a situation; we shall however give an example of this kind in 39.

Now let S be a subset of an arbitrary group G. B y Lemma 9.1 there is a unique minimal o)-subgroup of G containing S, namely the intersection of the w-subgroups of G containing S; we call this w-subgroup of G the o~-closure o / S in G and we shall denote it by cl~ (S, G) or b y cl~ (S), if there is no consequent ambiguity. I t is useful to have an alternative charac- terisation of cl. (S). This is provided b y Theorem 9.3.

THEOREM 9.3. Let S be a subset o] a group G. P u t S 1 = S and H 1 =gp(S1). Define Hi+ 1 inductively by putting

S~+1 = ( g i g ~ e H i , gEG, p e w }

and Hi+l = gp (Si+l).

T h e n c]~ (S) = 5 Hi.

i = l

Proo/. Let H * = 5 H~ and suppose g ' E H*, where g E G and p Ew. Thus g ' E H i for some

i = l

i and hence, b y definition, gESi+i and so g E H * . Consequently H* is an w-subgroup of G containing S and therefore

H* ~> cl~ (S). (9.31)

On the other hand, it is clear that cl~(S)>~ H 1. Suppose in fact t h a t cl~(S)>~ H~.

Obviously then clo (S)/> Si+ 1 and so cl~ (S) ~> H~+ 1. I t follows b y induction t h a t cl~ (S) >~ H j for all j and so

cl~ (S) >/H*. (9.32)

Putting (9.31) and (9.32) together we have the required result.

COROLLARY 9.4. The w-closure o/ a normal subset S o / a group G is a normal sub- group o/ G.

Proo/. We make use here of the representation of cl~(S) afforded b y Theorem 9.3;

consequently we adopt the notation used there. The proof is by induction. Suppose we have proved H i ~ G. Then Si+l is also normal in G. For let g E Si+l and x E G. Since gP E Hi ~ G we have x - l g P x = ( x - l g x ) ~ E H i and so

x-igx

E S~+ 1. Consequently Hi+ 1 ~ G. I t follows t h a t Hi <~ G for all i and so

cl~ (S) = 5 Hi <~ G.

1 = 1

I n a similar way one can prove that the w-closure of a characteristic subset is charac- teristic and that the w-closure of a fully invariant subset is fully invariant.

(8)

224 OrLBERT BAUMSLAG

10. If pEo~ and GE U~ then p t h roots are unique in G. Thus we m a y speak of " t h e "

p t h root of g E G whenever g has a p t h root. We shall sometimes denote the p t h root of g, if it exists, by g~.

I n U~-groups there is a certain interaction between elements and their p t h roots. The corollaries to the following lemma illustrate this interaction.

LEMMA 10.1. Let G, H E U~ and let 0 be a homomorphism o/ G into H. T h e n

(g~)O = (gO)~; (lO.11)

this is to be interpreted as stating that gO has a p-th root i / g does and in this case (10.11) holds.

Proo/. By applying 0 to the equation (g~)V = g we obtain (g~)'O = ((g~)OF = gO I t follows that gO has a pth root and t h a t

( ( g ~ ) 0 ) ' ~ = (g~)O = (gO)~.

COROLLARY 10.2. Let g, hEG, GE U~. T h e n g-l h z g = (g-l hg)z~.

Proo/. The mapping ~ defined b y x~ = g - l x g , where g is fixed and x ranges over the elements of G, is an automorphism; hence Lemma 10.1 applies and the result follows.

COROLLARY 10.3. (Kontorovi6) Suppose GEU~,, g, bEG, p, qE~o and that k and 1 are non-negative integers. Then g,k and h q~ are permutable i / a n d only i / g and h are permutable.

Proo/. We m a y assume k = 1, 1 = 0. Corollary 10.2 can now be applied:

g = g ~ = (h-lgPh)~ = h -1 (gPz)h = h - l g h ,

and so g and h are permutable if g~ and h are permutable. The converse is immediate.

COROLLARY 10.4. Let g, h be two permutable elements in a U~-group G. T h e n

(gh)Te = gT~h~; (10.41)

this is to be interpreted as stating that i / g h , g and h have p-th roots then (10.41) holds.

Proo/. Since g and h commute so do g z and h~ (by Corollary 10.3). Hence (gzchze) ~ = (g7~) p (h~) p =gh,

and the result follows on applying g to this equation.

(9)

SOME A S P E C T S OF G R O U P S W I T H U N I Q U E I~OOTS 2 2 5

11. W e shall a d o p t t h e following n o t a t i o n a l conventions: I n s t e a d of U(v}, E{v) a n d D(v) we shall w r i t e Up, E v a n d D v r e s p e c t i v e l y . A D2-grou p will be called a a - g r o u p a n d a Da-group a v-group. T h e s q u a r e r o o t of a n e l e m e n t g in a (~-group G will be d e n o t e d b y ga; t h e cube r o o t of a n e l e m e n t g in a T-group G will be d e n o t e d b y gT.

W e h a v e seen in C o r o l l a r y 10.4 t h a t if g a n d h a r e p e r m u t a b l e e l e m e n t s of a Dv-grou p t h e n (gh)~ = g ~ h z . T h e converse is n o t t r u e in general. H o w e v e r , we shall p r o v e t h i s converse in some special cases.

THEORE3~ 11.1. Two elements g and h in a (~-group G are permutable i/ and only i/

(gh)(~ = gc~h~.

Pro@ S u p p o s e (gh)a - g a h c r . S q u a r i n g b o t h sides of t h i s e q u a t i o n y i e l d s gh =gah(~gaha;

on c a n c e l l a t i o n of g a on t h e left a n d h a on t h e r i g h t t h i s e q u a t i o n r e d u c e s t o gah(~ =hag(~.

C o n s e q u e n t l y g h = (ga) ~ (ha) ~ = (ha) e (ga) e = hg.

On t h e o t h e r h a n d , if gh = hg t h e n (gh)a - g a h a , b y C o r o l l a r y 10.4.

C O r O L L A r Y 11.2. I n a a-group G the mapping which takes each element into its square root is an automorphism i] and only i / G is abelian.

W e n e e d t o digress for t h e m o m e n t t o r e c o r d a few p r o p e r t i e s , c o n n e c t e d w i t h e l e m e n t s of finite order, of U~-groups a n d D~-groups.

W e shall call a g r o u p G (o-free if i t does n o t c o n t a i n e l e m e n t s of o r d e r p if p e w . LEMMA 11.3. I / G E U , , then G is e)./ree.

Proo/. L e t g E G, p E co a n d s u p p o s e g~ = 1. T h e n gV = 1 = i v, a n d as G E U~ we h a v e g = 1.

L]~MMA 11.4. Let G be an extension O/a U~-group A by a Uo~-group B. L e t / , gEG have /inite order modulo A and suppose

F = gV (p

Eco).

(11.41)

Then / = g.

(10)

226 GILBERT BAUMSLAG

B y hypothesis there exists m prime to p such t h a t / m , gmEA ; hence (el. (11.41)) Proof.

(fm)~ = (gm)~

is an e q u a t i o n in the U~-group A a n d so

/m= gm. (11.42)

W e choose 2,/~ so t h a t 2 m + / z p = 1;

this is possible since (m, p) = 1. Then, m a k i n g use of (11.41) a n d (11.42) we h a v e g ~ ~r ~ g,tm gpp = f).m

fI~P

= f X m + / a p = f,

which completes the proof.

The following l e m m a is due, in part, to P. Hall.

THEOREM 11.5. Let G be an extension o / a Uo~-group A by a periodic m-free group B.

Then G 6 U~,. If, in addition, A E E~ then G E D~.

Proof. The first p a r t of the t h e o r e m is an i m m e d i a t e consequence of L e m m a 11.4;

t h u s we are left only with the second part. Suppose t h e n t h a t g E G a n d p Era. There exists a n integer m prime t o p such t h a t gmE A. Therefore we can find a E A such t h a t

gm = a T. (11.51)

Since A ~ G, g - l a g E A a n d so the resulting e q u a t i o n (cf. (11.51)) (g-lag)P = a T

is an e q u a t i o n involving o n l y elements of A, therefore

g - l a g = a. (11.52)

Since (m, p) = 1 we can choose 2,/~ such t h a t 2 m + # p = 1. T h e n (cf. (11.51) a n d (11.52)) g = g~m+,p = aa~ g , , = (a ~ g,),

a n d so g has a p t h root, a n d this completes the proof.

COROLLARY 11.6. A periodic group G is a D~-group i / a n d only if it is m-free.

Proof. I f G is m-free we can a p p l y T h e o r e m 11.5 to deduce t h a t G E D ~ - - w e m a y t a k e A = 1. On t h e other hand, if GED~ t h e n L e m m a 11.3 applies a n d G is w-free.

We m a k e use of Corollary 11.6 t o give an example of a situation in which

GeD,

a n d

(11)

SOME ASPECTS OS GROUPS WIT~r U~IQL'~ ~OOTS 227 g, h in G are such t h a t ( g h ) z = gJrh~ a l t h o u g h g h # hg. This is given in order to show t h a t

T h e o r e m 11.1 c a n n o t be genera]ised in the obvious way. W e t a k e G = g p ( a , b; a 4 = b ~ = 1, a - l b a =b3).

N o w ]G I = 20 a n d so b y Corollary 11.6 G is a z-group. I t can easily be verified t h a t (ab) 3 = a3b 3.

Hence (a3b3)T = (ab)3 T = a b = aa Tb3 v.

However, it follows f r o m the defining relations of G t h a t

a ~ b 3 # b 3 a s.

I n this example G is n o t even nilpotent; for locally nilpotent groups however t h e result for z-groups corresponding to T h e o r e m 11.1 does hold:

T ~ O R E M 11.7.

if a n d only i/

T w o elements g and h i n a locally nilpotent z-group G are permutable

(gh)T = g T h ~ . Proo/. If g h = hg t h e n (gh)~ = g T h T b y Corollary 10.4.

I t remains to show t h a t g h = h g whenever (gh)T = g v h v . To this end let H be a n y nilpotent subgroup of G containing g v a n d h T - - t h e existence of such a subgroup H is t a k e n care of b y T h e o r e m 15.1. T h e proof t h a t 9 a n d h c o m m u t e will be b y induction over the class c of H. I f c = 1 the result is immediate. I f c > 1 t h e n the f a c t o r g r o u p H / Z , with Z t h e centre of H, is a y-group of class c-1 (see e.g. Corollary 14.4). N o w

( g Z h Z ) ~ = ( g h Z ) T = ( g h ) ~ Z = ( g ~ h T ) Z = g T Z h ~ Z = ( g Z ) ~ ( h Z ) T . Thus inductively

g Z h Z = h Z g Z . consequently

g T Z h ~ Z = (gZ)~ ( h Z ) v = ( h Z ) ~ ( g Z ) T = h ~ Z g v Z .

Hence [hv, gT] = z EZ.

W e have (gh)T = g ~ h v , b y the hypothesis; on cubing this equation we obtain g h = g ~ h T g ~ h T g v h ~ : = (gT)2hT [hv, g v ] h T g T h v =

= (gT) 2 (h~)2gThTz (since z EZ) = (g~)a (hT)2 [(by)2, gv] h T z

= (q~)3 (h~)a [hv, gT]h~[h~, gT]z = g h z a.

15 -- 6 0 1 7 3 0 3 3 . A c t a mathematica. 104. I m p r i m 6 le 21 d 6 c e m b r e 1960

(11.71)

(12)

228 G I L B E R T B A U M S L A G

I t follows t h a t z a = 1.

B u t G E D ~ ; t h e r e f o r e z = 1. H e n c e g v h v = h T g v ( b y 11.71) a n d so g h = h g .

This c o m p l e t e s t h e p r o o f of t h e t h e o r e m .

I t follows from t h i s t h e o r e m t h a t if a l o c a l l y n i l p o t e n t T-group G has t h e p r o p e r t y t h a t t h e m a p p i n g w h i c h t a k e s e v e r y e l e m e n t i n t o its cube r o o t is a n a u t o m o r p h i s m , t h e n G is abelian. H o w e v e r , t h e r e s t r i c t i o n t h a t G be l o c a l l y n i l p o t e n t is r e d u n d a n t .

T H ~ o g ~ M 11.8. L e t G be a T-group i n which the m a p p i n g which takes every element into its cube root is a n a u t o m o r p h i s m . T h e n G is abelian.

Proo/. L e t g, h E G. T h e n

(gh)w = g z h v , (hg)T = hTgw.

T h e r e s u l t of c u b i n g b o t h sides of t h e e q u a t i o n (gh)T = g v h T a n d t h e n cancelling g v on t h e left a n d h v on t h e r i g h t is t h e f u r t h e r e q u a t i o n

(gv) ~ (h~) ~ = (h T gT) 2.

S t a r t i n g i n s t e a d f r o m t h e e q u a t i o n (hg)T = hTgT a n d p r o c e e d i n g s i m i l a r l y t h e c o r r e s p o n d i n g e q u a t i o n

(hT) ~ (gT) ~ = (gvhT) ~ c a n b e o b t a i n e d . H e n c e

(gw)e h = (g~)e (h T)3 = (gw)~ (h T)2h ~ = (h T gT)2 h v

= hT (gvhTg'~hw) = hT (gTh~c) e = hT (hT) ~ (gT) ~ = h (gT) 2.

T h u s h c o m m u t e s w i t h (gT) 2 a n d t h e r e f o r e so also does hT. C o n s e q u e n t l y (hTg~) 2 = (gT) ~ ( h T ) ~ = ( h ~ ) ~ ( g ~ ) ~ .

This l e a d s a f t e r c a n c e l l a t i o n of h~ on t h e left a n d gT o n t h e r i g h t t o g v h v = h ~ g T ,

a n d so g a n d h also c o m m u t e a n d G is abelian. This c o m p l e t e s t h e proof.

A n e x a m p l e of a n o n - a b e l i a n D r - g r o u p in which g a c t s as a n a u t o m o r p h i s m is t h e q u a t e r n i o n g r o u p Q of o r d e r 8:

Q = g p (a, b; a 4 = b 4 = 1, [a, b] = a 2 = b2).

(13)

SOME ASPECTS OF GaOUPS WITH VNIQUE ~OOTS 229 W e m a y t a k e p here t o be t h e p r i m e 5. T h e n

( e d ) ~ = c ~ d ~

for all c, d 6 Q since e a c h e l e m e n t in Q coincides w i t h its f i f t h root. This e x a m p l e shows T h e o r e m 11.8 c a n n o t be e x t e n d e d t o i n c l u d e all D r - g r o u p s .

CHAPTER II

Locally nilpotent groups, ZA-groups and nilpotent groups

12. W e shall be c o n c e r n e d here w i t h g r o u p s s a t i s f y i n g c e r t a i n c o m m u t a t i v i t y condi- tions. F o r e x a m p l e we p r o v e t h a t t h e t e r m s of t h e u p p e r c e n t r a l series of a Z A - g r o u p i n E(o a r e og-ideals. T h e m a i n r e s u l t in t h i s c h a p t e r is of a d i f f e r e n t k i n d : T h e og-closure of a n i l p o t e n t s u b g r o u p of class c of a Uo~-group is n i l p o t e n t of class c. W e d e d u c e a n u m b e r of r e l a t e d r e s u l t s a n d , in p a r t i c u l a r , a s o r t of d u a l t o t h e r e s u l t m e n t i o n e d a t t h e o u t s e t : T h e first o9 t e r m s of t h e lower c e n t r a l series of a l o c a l l y n i l p o t e n t D ~ - g r o u p are og-ideals.

S o m e of t h e r e s u l t s s t a t e d , a n d s o m e t i m e s p r o v e d here, are, p e r se, similar t o k n o w n r e s u l t s for R - g r o u p s a n d d i v i s i b l e groups; in p a r t i c u l a r t h e r e is a c e r t a i n a m o u n t of over- l a p p i n g b e t w e e n our r e s u l t s a n d t h o s e of K o n t o r o v i ~ [18] a n d C e r n i k o v [7].

13. A set of g r o u p s {H~}, w h e r e i r a n g e s o v e r a w e l l - o r d e r e d i n d e x set 1, is said t o f o r m a n ascending sequence if H~ < H j w h e n e v e r i < ]. S u p p o s e n o w t h a t e a c h H~ is a o9-subgroup of some f i x e d supergroup(1) G a n d l e t H = LIH~. T h e n H is itself a ~o-subgroup of G. F o r

i E I

if g e G, p co9 a n d g~ 6 H , t h e n g~ 6 H~ for s o m e i 61. C o n s e q u e n t l y g e H~ a n d so g 6 H . T h u s we h a v e p r o v e d

LEMMA 13.1. The union o/ an ascending sequence o/ og-subgroups o/ an arbitrary group is an og-subgroup.

I n a s i m i l a r w a y we can p r o v e :

LEMMA 13.2. The union o/ an ascending sequence of og-ideals of an arbitrary group is an og-ideal.

T h e following t h e o r e m a n d C o r o l l a r y 13.4 a r e i m m e d i a t e g e n e r a l i s a t i o n s of a t h e o r e m of K o n t o r o v i ~ [18] on R - g r o u p s - - t h e proofs a r e s i m i l a r a n d a r e t h e r e f o r e o m i t t e d .

THEOlCEZ~ 13.3. The centraliser o/ an arbitrary set of elements o/ a U~o-group is an og-subgroup.

(1) G is a supergroup of H~ if H~ ~< G.

(14)

230 G I L B E R T B A U M S L A G

COROLLARY 13.4. The terms o/the upper central series of a U~-group are m-ideals.

COROLLARY 13.5. A n w-subgroup H o / a U~-group G which is contained in the centre is an w-ideal o/G.

Proo/. Suppose p e w , x, y E G and (xH) p = (yH) p. Then

xP=yPh, hEH. (13.51)

The centre ~(G)is an co-ideal of G (Corollary 13.4); furthermore ~(G) contains H. Conse- quently (x ~ (G)) p = (y~ (G)) p and so x~ (G) = y~ (G). Thus x = yz, z E ~ (G) and

x p = yPz p. (13.52)

On comparing (13.51) and (13.52) we see t h a t h = z p. B u t as H is an w-subgroup of G, z E H.

Consequently x H = y H and H is therefore an m-ideal of G.

We saw in 9 t h a t a normal w-subgroup of a group is not necessarily an ~o-ideal; however, this is always true whenever the factor group is locally nilpotent. This follows immediately from the following theorem, which is due to Mal'cev [26] and Cernikov (see Kurosh [21]

vol. 2, p. 247).

THEOREM 13.6. A locally nilpotent group is a U~-group i / a n d only i / i t is co-/ree.

COROLLARY 13.7. A normal w-subgroup N o / a group G is an o).ideal o / G i / G / N is locally nilpotent.

Proo/. B y Theorem 13.6 G / N e U~, since it is m-free.

14, We shall consider in this section some properties of various kinds of E~-groups.

First we prove:

THEOREM 14.1. Let G be a ZA-group, let p e w and let G be an E~-group. Then the set o/ elements o/ G whose orders are a power o / p / o r m s a subgroup contained in the centre

olG.

The proof of Theorem 14.1 depends on the following lemma (cf. Kurosh [21] vol. 2, p. 234).

LEMMA 14.2. Let G be a ZA-group in the class E~. Then the centre Z =~(G) is an co-ideal o/G.

Proo/. L e t 1 =# geG, let peru and suppose gPeZ. Choose a so that(l) (1) We define ~1 (G) = ~ (G),

by ~ (G/~ ~ (G)) = ~a+l (G)/~a (G).

~a (G)= [.J ~ (G) if ~ has no predecessor, and ~+1

~<~

(15)

SOME A S P E C T S OF G R O U P S W I T H U N I Q U E R O O T S 231

gr gE~+~(G).

I f ~ = 0, t h e n gEZ, as r e q u i r e d . Suppose, if possible, t h a t ~ = 1 a n d l e t x be a n a r b i t r a r y e l e m e n t of G. L e t x 0 b e a p t h r o o t of x. T h e n

1 = [g', x] = [g, x3] = [g, x]

a n d so a ~ 1. L e t us n o w s u p p o s e t h a t a n y e l e m e n t of $~ (G) w h i c h has a p t h p o w e r in Z, for e v e r y fi s a t i s f y i n g 1 ~ f i ~< ~, is c o n t a i n e d in Z. T h e n

[x, g] = x lg-lx.g e ~ (G),

a n d as b o t h g a n d x - l g - l x h a v e o r d e r p m o d u l o Z, [x, g] itself h a s o r d e r m o d u l o Z a p o w e r of p(1) a n d hence belongs t o Z. This h o l d s for all xEG a n d so g E ~ ( G ) , which, as we s h o w e d a b o v e , i m p l i e s g EZ. This c o m p l e t e s t h e p r o o f since a n o r m a l co-subgroup of a l o c a l l y nil- p o t e n t g r o u p is a n co-ideal (Corollary 13.7).

T h e p r o o f of T h e o r e m 14.1 n o w follows easily. F o r if g~" = 1 t h e n gP'E~(G), a n d so, b y L e m m a 14.2, g E ~ (G).

T h e first of t h e following t w o corollaries is d u e t o ~ e r n i k o v [1]; t h e p r o o f of T h e o r e m 14.1 is b a s e d on his original proof.

COROLLARY 14.3. In a divisible ZA-group the elements o//inite order/orm a subgroup o/the centre.

COROLLARY 14.4. The terms o/the upper central series o / a ZA-group G in E~ are co-ideals.

Proo/. B y C o r o l l a r y 14.2 Z = ~ (G) is a w-ideal. T h u s G/Z is a U~-group a n d so b y C o r o l l a r y 13.4 t h e t e r m s ~ (G)/Z of t h e u p p e r c e n t r a l series of G/Z are co-ideals of G/Z. I t follows e a s i l y t h a t t h e t e r m s ~ (G) of t h e u p p e r c e n t r a l series of G are co-ideals of G.

W e r e m a r k t h a t t h e c e n t r e of a l o c a l l y n i l p o t e n t E ~ - g r o u p is n o t n e c e s s a r i l y a n co-ideal since e v e r y l o c a l l y n i l p o t e n t p - g r o u p can b e e m b e d d e d in a d i v i s i b l e l o c a l l y n i l p o t e n t p - g r o u p w i t h a t r i v i a l c e n t r e (see B a u m s l a g [3]) (cf. C o r o l l a r y 14.3).

T H E O R E M 14.5. The terms o/the lower central series o / a nilpotent group G in E~ are themselves E~-groups.

(1) In a locally nilpotent group the elements of order a power of p form a subgroup (see Kurosh [21] vol. 2, p. 215).

(16)

232 G I L B E R T B A U M S L A G

Proo 1. The proof is b y induction on the class c of G. I n the case c = 1 the result is im- mediate, so we have a basis for induction. Now Fc(G) is generated by the commutators:

[g,h], gCG, heF~_I(G).

If pEco then g has a pth root, say gu. Hence

[g, h] = [gg, h] = [go, h]"

and so [g, h] has a pth root in Fc (G). Consequently every element of the abelian group Fc (G) has a p t h root and therefore Fc (G)E E~. Consider now the factor group H = G/F~ (G).

Then H E E~ and is nilpotent of class c - 1 and we m a y apply induction to deduce t h a t F ~ ( H ) = F ~ ( G ) / F ~ ( G ) is an E~-group for i = l , 2 . . . c. I t follows immediately t h a t 1~ (G)E E~ for i = 1, 2 . . . c and the theorem is proved.

I t is easy to make examples of nilpotent E~-groups in which the terms of the lower central series are not co-ideals. On the other hand, for nilpotent D~-groups such examples cannot exist.

COROLLARY 14.6. The terms o / t h e lower central series o / a nilpotent group G in D~

are co-ideals.

Proo/. A normal co-subgroup of a (locally) nilpotent U~-group is an co-ideal (Corollary 13.7) and so the result follows from Theorem 14.5, because an E~-group which is a sub- group of a U~-group is an co-subgroup of t h a t group.

B y Corollary 14.6 the derived group of a nilpotent D~-group is an co-ideal. This result is, however, not true for D~-groups in general. An example of a D~-group in which the

derived group is not an co-ideal will be given in Part I I (see Theorem 37.2).

If we consider the more general class U~, then again Corollary 14.6 is no longer true.

For let

G = g p ( a , b, c; [a, b] = c 2, ca = a c , cb = b c ) .

I t is easy to verify t h a t GE U s and t h a t G is nilpotent of class two; but the derived group of G is generated b y c 2 and so is not an co-ideal of G.

15. Mal'ccv [25] has proved t h a t if G is a torsion-free locally nilpotent group then G can be embedded in a torsion-flee locally nilpotent divisible group G*. We shall prove here, as an application of the main theorem of this section, t h a t if G is a torsion-flee locally nilpotent group and if K is a supergroup of G, which is a divisible R-group, then G*, the

(17)

S O M E A S P E C T S O F G R O U I " S W I T H U N I Q U E R O O T S 2 3 3

0-closure of G in K (~ being t h e set of all primes) has no o p t i o n o t h e r t h a n t o be a l o c a l l y n i l p o t e n t d i v i s i b l e g r o u p .

T h e m a i n t h e o r e m is t h e following.

TH]~OnEM 15.1. Let G be a subgroup o/ a U~o-group K. Then, i / G is nilpotent o] class c, so is cl~ (G, K).

Proo/. L e t

1 < Z 1 < Z 2 < " " < Z c = G

be t h e u p p e r c e n t r a l series of G. I t follows, on a p p l y i n g T h e o r e m 13.3 that(1) [cl(Z)~ G] = 1,

a n d s i m i l a r l y t h a t [cl(G), el(Z)] = 1. (15.11)

T h u s we see f r o m (15. i 1) t h a t cl (Z) is a s u b g r o u p of t h e c e n t r e of cl (G) a n d so, b y C o r o l l a r y 13.5, cl(Z) is a n o)-ideal of cl(G); we shall m a k e use of t h i s f a c t in t h e sequel.

N o w if c = 1 t h e n , b y (15.11), [cl(G), cl(G)] = 1 a n d so cl(G) is a b e l i a n - - t h u s we h a v e p r o v e d t h e t h e o r e m in t h e case c = 1. W e shall use t h i s f a c t as t h e basis for a n i n d u c t i o n : if L is a s u b g r o u p of a U~o-group M a n d if L is n i l p o t c n t of class d ( d < ~ c - 1) we shall a s s u m e cl~ (L, M ) is n i l p o t e n t of class d.

W e n o w m a k e use of T h e o r e m 9.3. W e p u t

S 1 = e " c l ( Z l ) ;

a c c o r d i n g t o T h e o r e m 9.3 if H 1 = gp(S1), S~+ 1 = {g[gVEH~, gEK, p e w } , H~+I = gp(S~+l),

r162

t h e n cl(S1) = U1H~. W e shall s h o w t h a t H~ is n i l p o t e n t of class e for i = 1, 2, ... a n d conse- q u e n t l y t h a t c l ( G . c l ( Z x ) ) is n i l p o t e n t of class c. I t follows f r o m (15.11) t h a t $1 is a sub- group; t h e r e f o r e H x = Gcl(Z1). N o w

H1/cl (ZI) ~- ~ / G n el (Z~) = G/Z~;

t h u s H 1 is n i l p o t e n t of class c since cl (Z1) ~< ~ (H1).

S u p p o s e now t h a t we h a v e p r o v e d H~ n i l p o t e n t of class c for all i ~ ], for some ] ~> 1.

W e n o w w e l l - o r d e r t h e e l e m e n t s of Sj+I:

Sj+ 1 ~ {al, a 2 , . . . , a . . . ).

(1) Throughout the proof of this theorem we sha.ll simply write el (H) for the o)-closure of H in K.

(18)

234 G I L B E R T B A U M S L A G

D e f i n e F ~ = g p ( H ] , al, a s . . . a~ . . . . ; ~ </~).

N o w F 1 is n i l p o t e n t of class c. To see t h i s we consider HJcl(Z1) as a n i l p o t e n t s u b g r o u p (of class c - 1) of F1/cl(Z1). T h e r e is a p r i m e p e w for w h i c h (a 1 cl (Z))P e H J c l (Z1) a n d so, b y i n d u c t i o n , F1/cl (Z1) = cl ( H J c l (Z1), F 1 / c l (Z~)) is n i l p o t e n t of class c - 1. B u t cl (Z~) ~< $ (F1) a n d so F 1 is n i l p o t e n t of class c. I t follows b y t r a n s f i n i t e i n d u c t i o n t h a t

Hi+l :

U

F~

is t h e u n i o n of a n a s c e n d i n g sequence of s u b g r o u p s , e a c h n i l p o t e n t of class c a n d so Hi+ 1 is n i l p o t e n t of class c. Therefore, b y i n d u c t i o n , e a c h H t is n i l p o t e n t of class c a n d hence e l ( G . c l ( Z ) ) = U H~ is n i l p o t e n t of class c. I n p a r t i c u l a r el(G) is n i l p o t e n t of class c a n d

i = l

t h i s c o m p l e t e s t h e p r o o f of t h e t h e o r e m .

COROLLARY 15.2. Let G be a subgroup o / a U~-group K. I] G is locally nilpotent then its co-closure in K is also locally nilpotent.

Proo]. L e t ~ , g2 . . . ~ be n e l e m e n t s t a k e n a r b i t r a r i l y f r o m cl (G) a n d l e t

H = g p ( ~ , ~2 . . . . , ~ ) .

W e h a v e t o show t h a t H is n i l p o t e n t . W e m a k e use of T h e o r e m 9.3. a n d w r i t e el (G) = U H i ,

t = l

w h e r e H 1 = G x = G,

H t + l = gp (G~+l)

a n d G i+l = {gig 6 K, g~ 6Hi for s o m e p 6co).

T h i s e x p r e s s i o n for el (G) e n a b l e s us t o f i n d a f i n i t e set

such t h a t H ~< cl((g~, g2 . . . g~)).

To see t h i s let us consider t h e case of a single e l e m e n t g E cl (G). Since g belongs t o t h e u n i o n of t h e H i i t m u s t belong t o one of t h e m , s a y t o H i . T h e r e f o r e we can w r i t e

g = hlh 2 ... h~,

(19)

SOME A S P E C T S OF G R O U P S W I T H U N I Q U E ROOTS

where h~ C Gj and has therefore a prime p , Coo associated with it such that

235

h~, EHj_I.

Now, inductively, corresponding to each element in Hi_ 1 we can find a finite set of elements in G whose og-closure contains that element. Thus we can find

such t h a t ]~Ptt (~cl({g/~,l ' gtt, 2, . . . , gtt, nit}).

Then, since p~Eeo, we have also

~ . e cl({g.,1, g.,2, ...,

g., n.)).

Hence gEcl(gl,1, gl, s . . . g~,n~).

I t follows that we can, in a similar way, find a set

such that

= {gl, g2 . . . . , gin} (gi e G),

~ , e c l ( g ) ( i = 1 , 2 . . . . ,n};

consequently H < cl (0).

The group generated b y a finite subset ~ of a locally nilpotent group G is necessarily nil- potent. Furthermore, b y Theorem 15.1, the co-closure of a nilpotent subgroup of a U,~- group is again nilpotent. Now el(G ) = cl (gp(O)); thus H is a subgroup of a nilpotent group and so is itself nilpotent. This completes the proof of the corollary.

COROLLARY 15.3. Let K E U,~, let G be nilpotent o/class c and suppose G is a subgroup o / K . Suppose, /urthermore, that every element k E K has a power k n E G, where all the prime divisors o] n belong to r T h e n K is nilpotent o/class c.

Proo/. The to-closure of G is K , so the theorem follows on applying Theorem 15.1.

I t m a y be of interest to note a connection between Corollary 15.3 and a theorem of Duguid and Maclane [8] which states t h a t a subgroup of finite index in a torsion-free nilpotent group of class c is also of class e. Since a torsion-free nilpotent group is an R-group (Theorem 13.6) their theorem is an immediate consequence of Corollary 15.3.

Corollary 15.3 has an analogue for locally nilpotent groups, for it follows easily from Corollary 15.2 that ff K E U~ and ff G is a locally nilpotent subgroup of K such t h a t every

(20)

236 G I L B E R T B A U M S L A G

]c E K has a power /c~E G, where all t h e prime divisors of n belong to (o, t h e n K is itself locally nilpotent. One m a y also prove a similar result for Z A - g r o u p s .

T H E O R E m 15.4. The /irst eo terms o/ the lower central series o/ a locally nilpotent D~.group G are m-ideals o / G .

Proo/. L e t

G = H o >~ H I >~ . . . >~ H ~ >~ . . . >~ I

be t h e lower central series of G. Suppose p e w , g E G a n d g~EH~ for some finite integer i;

t h e n g~ can be written as a p r o d u c t of (say) n / - f o l d c o m m u t a t o r s involving t h e n ( i + 1) elements

gl,1, gl, 2, ..., g1.~+1, ..., gn. i+l;

a 1-fold c o m m u t a t o r is a simple c o m m u t a t o r [x, y] and, inductively, a n / - f o l d c o m m u t a t o r is the c o m m u t a t o r of a n (i - 1)-fold c o m m u t a t o r a n d a n element of G. We p u t

H = gp (gl. 1, gl. 2 .. . . . g~. i + l ) "

T h e n H is nilpotent a n d so therefore el(H) is also nilpotent, b y Theorem 15.1. The terms of t h e lower central series of cl (H) are ~o-ideals, b y Corollary 14.6; t h u s since gP belongs t o the i t h m e m b e r of t h e lower central series of cl (H) (by t h e choice of H), g belongs also t o t h e ith m e m b e r of the lower central series of cl(H). Consequently g EH~ a n d hence H~ is an w-subgroup of G. F u r t h e r m o r e H~ is n o r m a l in G a n d so on applying Corollary 13.7 we see t h a t H , is an ~o-ideal of G. The same analysis holds for e v e r y i a n d so we have H , is an

oo

co-ideal of G for i = 1, 2, . . . . Hence H~ = ['IH~ is also an w-ideal of G ( L e m m a 9.2). This

i = l

completes the proof of the theorem.

I do n o t k n o w w h e t h e r it is possible to e x t e n d this t h e o r e m so as to include all t h e members of t h e lower central series of a locally nilpotent Due-group.

COROLLARY 15.5. The terms o / t h e derived series o/ a locally nilpotent D~-group are w-ideals.

Proo/. B y L c m m a 9.2 t h e intersection of o-ideals is an w-ideal a n d the corollary follows i m m e d i a t e l y f r o m this r e m a r k a n d T h e o r e m 15.4.

We complete this section with t h e proof of t h e following t h e o r e m a n d its corollary.

TtIEOREM 15.6. Let G be a nilpotent group and suppose that G is generated by its p-th p o w e r s , / o r each prime p E w . Then G is an E~-group.

(21)

P r o @ S u p p o s e

S O M E A S P E C T S O F G R O U P S W I T I - I U N I Q U E R O O T S

G = Ho > H I > . . . > H~ = I

237

is t h e lower c e n t r a l series of G. T h e t h e o r e m is o b v i o u s l y t r u e w h e n c = 1; we shall use t h i s f a c t as a basis for a n i n d u c t i o n o v e r t h e class c of G.

W e p r o v e first t h a t H~_ 1 is a n E ~ - g r o u p . T h e e l e m e n t s of He_ 1 can be w r i t t e n as p r o d u c t s of c o m m u t a t o r s of t h e f o r m

h = [g, g'], g e G , g'eH~_2.

~ o w G is g e n e r a t e d b y i t s p t h p o w e r s a n d so g can b e w r i t t e n in t h e f o r m

(15.61)

p p

g = g l g ~ ... g~,

where t h e g~ are e l e m e n t s of G. W e s u b s t i t u t e t h i s e x p r e s s i o n for g in e q u a t i o n (15.61) a n d m a k e use of t h e e q u a t i o n

[lg, h] = [1, h] ~ It, h]

t o show t h a t h is a p t h p o w e r of a n e l e m e n t in H~_ 1. E x p l i c i t l y

P P . . P p P

h = [g~g~ ... g~, g'] = [g~, g,]g2 % " % . [g2 g~ ... g~, g']

= [gL g'] [g~,~ g,] ~ 3 ~ ~n. [ g ~ . . . g~, g,] = I t , g,]~ [g2," g'] . . . =

= I t , g']~ [g~, g']~ ... [g~, g ' F = f i g , g'] [g~, g'] ... [g~, g ' ] ] Y .

T h u s h is a p t h p o w e r of a n e l e m e n t in He_ 1 a n d i t follows t h a t e v e r y c l e m e n t of He_ 1 is a p t h p o w e r of a n c l e m e n t in H c - r T h i s p r o c c d u r c a p p l i e s for all p e w a n d since Hc-x is abelian, i t is a n E ~ - g r o u p . N o w G is g e n e r a t e d b y its p t h p o w e r s for each p in ~o; t h e r e f o r e so also is G/Hc_I. N o w G/Ho_ 1 is n i l p o t c n t of class c - 1 a n d so we arc a b l e t o a p p l y t h e i n d u c t i o n h y p o t h e s i s t o a s s e r t t h a t G/Hc_ 1 is in f a c t a n E ~ - g r o u p . T h u s G is a c e n t r a l e x t e n s i o n of a n E ~ - g r o u p b y a n Eo~-group a n d so b y T h e o r e m 21.2 (see C h a p t e r IV) G is itself a n Er This c o m p l e t e s t h e p r o o f of t h e t h e o r e m .

CHAPTER I I I

Construction of U~-groups, E~-groups and D~-groups

16. I n t h i s c h a p t e r we shall m a k e use of t h e / r e e product a n d t h e second nilpotent pro- duct. W e shall l a t e r also r e q u i r e t h e / t e e product with an amalgamated subgroup, a n d , m o r e g e n e r a l l y , t h e generalised/ree product. I t is c o n v e n i e n t t h e r e f o r e t o give here a s h o r t exposi-

(22)

238 G I L B E R T B A U M S L A G

tion on such products and to record some of their properties which w e shall need in sub- sequent work. As far as generalised free products are concerned, we shall take this op- p o r t u n i t y to draw heavily from B. H. N e u m a n n ' s "An essay on free products of groups with amalgamations"; we even go so far as to take the liberty of quoting from this essay (B. H. N e u m a n n [27]).

L e t F be a group and let F~ be subgroups of F, where ~ ranges over an index set A.

We call F the

generalised/ree product o/the F~ if

i) F is generated b y its subgroups F~, and ii) for every group G and every set of homomorphic mappings ~ of each F~ into G, every two W~, ~, of which agree where both are defined, there exists a homomorphic m a p - ping ~ of F into G t h a t coincides with ~ on each Fa (see B. H. N e u m a n n [27], Theorem 1.1). Now suppose # is the generalised free product of its subgroups F~(AEA) and p u t

F~nF,,=H~,,(=H,.a), ~,#eA.

I f all the intersections H~, coincide to form a single subgroup H:

F~n F~ =H, (;t. ~)

then F is called

the (generalised) /ree product o/the F~ with au amalgamated subgroup H.

I n the case where H = 1, the trivial subgroup, then F is called simply t h e / t e e

product,

or, to emphasise the distinction,

the ordinary/ree product o/the Fa.

Let now groups F~ be given, where ~ runs over a suitable n o n - e m p t y index set A.

I n every F~ and to every index

~eA

let a subgroup HA~ be distinguished; H n is always to be the whole group F~. I f there exists a group F which is the generalised free product of groups F~ with intersections

~%. = F~

n ~',,

= H ~ , and if there are isomorphic mappings ~z of F~ onto ~ ,

such t h a t always / ~ = H~.,~

t h e n we say t h a t

the generalised /tee product o/the F~ with amalgamated H~t, exists,

or simply the generalised free product of the F~ exists. The generalised free product does not always exist; however, in the special case of the generalised free product with a single subgroup amalgamated, the generalised free product does always exist (Schreier [32]).

I t is often convenient when dealing with generalised free products not to distinguish

(23)

SOME A S P E C T S OF G R O U P S ~'*ITI-I U ~ I Q U E R O O T S 239 between a group and an isomorphic copy of it; we shall adopt this convention whenever it is both convenient and unambiguous.

Let us suppose for the moment t h a t F is the free product of the groups F~ with amalgamated subgroup H (2EA). The elements in 2' can be represented b y a certain normal/orm: We choose in every group F~ a system S~ of left coset representatives modulo H containing the unit element; thus every element ]E F~ can be uniquely represented in the form

] = sh (sES~, hEH).

Now we distinguish certain words in elements of the F~; specifically we call

W ~ 8 1 8 2 . . . snh a normal word if it satisfies the following three conditions:

(i) E v e r y component s~(1 ~ i ~ n) is a representative ~ 1 belonging to one of the S~.

(ii) Successive components s~ belong to different systems of representatives; in other words, if 1 <~ i <~ n, s~ES~, s~+IES., , then ~4=/~.

(iii) The last component belongs to the common subgroup h EH.

We call n the length of the normal word. Note t h a t a word is a string of symbols; if we interpret it as a product (which is written in the same way) we obtain an element of the group, and we say the word represents the element. Then every element is represented by one and only one normal word (ef. B. H. Neumann [21] Theorem 2.4). The uniquely determined normal word representing / we call the normal/orm o / / a n d we call the length of the normal word representing / the length o / t h e dement /; we write ~ ( / ) - n if ] is of length n. The following lemma is due to B. It. Neumann [27]:

LEMMA 16.1. I / n > l, i/

/ =/d~ .../~,

and i / n o two successive/actors/~,/~+1 are elements o/the same group F~, then n i8 the length o//.

I / n = 1, the length o / / i s 0 or 1 according as / lies in H or not.

We call the element /E F cyclically reduced if none of its conjugates in F has smaller length t h a n itself. The following ]emma is due to B. H. ~Teumann [27].

LEMMA 16.2. I / / is cyclically reduced and i/ it has the normal/orm

/ ~ Sl8~. . . . 8 n h

o/ length n > 1, then s 1 and sn belong to different groups F~ =~ F , .

(24)

240 G I L B E R T B A U M S L A G

L E ~ ) ~ A 16.3. (B. H . N e u m a n n [27].) I] / has length n > 1 and i/ in its normal/orm

/ = s i s 2 . . ,

snh

the components s 1 and sn belong to di//erent groups F~ and YF~, then / is cyclically reduced.

T H E O R E ~ 16.4. (B. H . N e u m a n n [27].)

]Let F be the ]ree product o/ groups F~ with amalgamated subgroup H; i/ / is an element o/

/inite order in _Y, then / is conjugate to an element in (at least) one o/ the F~.

COROLLARY 16.5. (B. H . N e u m a n n [27].) The/ree product o/ locally in/inite groups with an amalgamated subgroup is locally in/inite.

Suppose n o w t h a t F is a group g e n e r a t e d b y its subgroups F~, where 2 ranges over a n ordered index set A. T h e n F is a regular product of the groups F~ (Golovin [11], [12]) if e v e r y element /E F has a unique regular representation of the f o r m

/ = ])*(1) ]).(2) "'" /.~(n) U,

where h(~) E F~r )~ (1) < ~t (2) < .-. < ~t (n), u E n m ([F~]) a n d [F~.]=gp([/~, /s]; h E F t , / ~ E F s, ~ # , ,~,

#EA).

W e r e v e r t to the case where F is t h e free p r o d u c t of t h e groups F~, 2 E A , A a n ordered index set. L e t

I[F~] = n m ([F~]), k[F~] = [F, k-l[F~]], where/c = 2, 3 . . . T h e n Golovin [ l l ] calls the f a c t o r group

F/~ [F~]

t h e Ic-th nilpotent product of t h e groups F~, A EA; f u r t h e r m o r e he p r o v e s t h a t the k t h nil- p o t e n t p r o d u c t of groups is in fact a regular product.

17. W e consider n o w m e t h o d s of constructing 'new' Uo)-groups, E~-groups a n d D~- groups f r o m given U~-groups, E~-groups a n d D~-groups.

T h e proof of the following l e m m a is s t r a i g h t f o r w a r d a n d is omitted.

:L~MMA 17.1. The restricted direct product and the unrestricted direct product o/ X~,- groups is an X~-group where X here stands/or any o/ the three letters U, E and D.

(25)

SOME ASPECTS OF GROUPS W I T H U N I Q U E ROOTS 2 4 1

K o n t o r o v i 5 [8] has p r o v e d t h a t t h e free p r o d u c t of R - g r o u p s is a n R-group; we genera]ise this result to the following

T ~ E O R E M 17.2. The/ree product F o/ Uo)-groups F~., where ~ ranges over an index set A, is a U~-group.

Proo/. Suppose peso,/, g C F a n d

/" = g ~ ( / .

1).

(17.21)

W e m a y assume / is cyclically reduced; t h e n the proof t h a t / = g falls n a t u r a l l y into two parts:

(i) 2(/) = 1. W e h a v e g-1]pg =]~ a n d s i n c e / @ 1 , / P + 1 a n d so 1 =,: (g-~Fg) =,~ (g~) >~ ,~ (g).

T h u s ~ ( g ) d l a n d therefore, b y (17.21), A(g) is precisely 1. I t follows t h e n with t h e aid of (17.21) t h a t / a n d g belong to the s a m e subgroup, say to F2.. B u t F~. is a U,o-group a n d therefore (17.21) yields / = g.

(ii) 2 (/) > 1. L e t us suppose t h a t the n o r m a l f o r m for / is

/ is cyclically reduced a n d consequently ~ ( 1 ) 4 )~ (m) ( L e m m a 16.2); therefore the n o r m a l f o r m o f / ~ is

/~ = A(I>/~(~) . . . A~,,,~ I~<:) -.- &m>.

Now, b y (i), t h e length of g is a t least 2. L e t t h e n

g = g,(l) g,(2) . . . g , Cn) ( n > 1)

be the n o r m a l f o r m of g. W e assert t h a t g is itself cyclically reduced. F o r it follows f r o m (17.21) t h a t g a n d / 9 are p e r m u t a b l e i.e.

g,(1) g/~(2) -.. gl~(~) I~(1)/~(2) ..- /~(,~) = A(~)/~(2) --- /~(~) g~(~) g~(2) .-- gi~(~). (17.22)

r a p m p

I f / ~ (n) 4= ~ (1) a n d # (1) 4= 2 (m) t h e n we see i m m e d i a t e l y f r o m (17.22) t h a t # (1) = 2 (1) a n d ff (n) - ~ (m) a n d so g is cyclically reduced since ~ (1) 4 ~ (m) ( L e m m a 16.3). I f # (n) 4= ~ (1) a n d #(1) = A(m) t h e n

X (g~,(~) g,(2> . . . g , ( ~ ) / ~ ( : ) h(2) . . . /~(ra)) = n + p m > ~ (h(~) A(~) ..- A(ra) g , ( : ) . . . g,<~)),

(26)

242 G~BERT BAVMSLAG

which is i n c o m p a t i b l e w i t h (17.22) a n d so this case does not arise. F i n a l l y if # ( n ) = ~(1) a n d ~u (1) = 2 (m) t h e n we have, once more, g cyclically reduced. H e n c e we m u s t always h a v e

g~ = g,(1) g~(~) ... g~(n) g~(1) ... g~(~),

which is t h e n o r m a l f o r m of gP; a n d so it follows f r o m (17.21) t h a t ] a n d g are in fact identical.

This completes t h e proof.

I t is clear t h a t t h e flee p r o d u c t of m o r e t h a n one n o n t r i v i a l E~-group is not a n E~- group since e x t r a c t i o n of p t h roots is n o t always possible. One m a y a s k w h e t h e r or n o t certain regular p r o d u c t s of U~-groups are again U~-groups and, in particular, w h e t h e r such regular p r o d u c t s w h e n applied t o D~-groups result in D~-groups.

Lv, MMx 17.3. The second nilpotent product F o/ U~,-groups F ~ ( ~ e A ) is a U~,.group i / a n d only i / [ F ~ ] / s w-]tee.

Proo/. I f [F~] contains elements of order p t h e n F is n o t a U~-group. Suppose t h e n t h a t this is n o t the case. L e t p e w a n d l e t / , 9 E F be such t h a t

F = g'. (17.31)

L e t / =/x(1)/~(e) ... /~(n) u, 2 (1) < 2 (2) < .-. < 2 (n), u E n m ([F~]) (17.32) g=g~a)g~(2)...g~(m)V, j u ( 1 ) < j u ( 2 ) < . . . < # ( m ) , v e n m ( [ F ~ ] ) (17.33) be t h e regular r e p r e s e n t a t i o n s for / a n d g respectively, where/~(~) e F~ (~) a n d g~(j)e F , (j).

I n the second nilpotent p r o d u c t [F~] lies in the centre (cf. Golovin [12]) a n d so in this case nm([F~]) = [F~]. T h u s we c a n easily deduce t h e regular r e p r e s e n t a t i o n s for ]P a n d gV f r o m those of / a n d g (cf. Golovin [12]):

] P = / ~ ( 1 ) / ~ ( 2 ) " ' ' ]~(n)" u P " ( 1~ [ / ~ ( i ) , /),(])])~9(p-1) (17.34)

l ~ i < j ~ n

g P - - -g,a)g,(2) .. g,(m) v "( I~ P P . P . P [g,(~>, g,(j)])89 (17.35) N o w b y (17.31)/~ a n d gP are equal; hence t h e regular representations (17.34) a n d (17.35) coincide. T h u s m = n and, in p a r t i c u l a r

P __ P

/)~(1) - - g/~(1) . . . /~(m) = g~(m).

B u t each Fa is a U~-group a n d therefore we h a v e

h ( i ) = g.(1), h(2) = g~,(~) . . . /~(,,) = g . , m ; .

(27)

S O M E A S P E C T S O F G R O U P S VVITtt U N I Q U E R O O T S 243 I t follows t h e n c e t h a t u p = v ~.

N o w u a n d v b o t h lie in t h e c e n t r e of F ; t h e r e f o r e (uv-1) ~ = 1, i.e. u = v a n d so (cf. 17.32) a n d (17.33)) / = g. This c o m p l e t e s t h e proof.

W e give n o w a n e x a m p l e of a s e c o n d n i l p o t e n t p r o d u c t of U~-groups w h i c h is n o t a U~-group. W e t a k e

G = g p ( a , b, c; [a, b] = c ~, [a, c] = [b, c] = 1), H = g p (d, e , / ; [d, e] = / 3 , [d, ]] = [ e , / ] = 1).

I t is e a s y t o show t h a t G a n d H a r e U3-groups. B u t F , t h e s e c o n d n i l p o t e n t p r o d u c t of G a n d H , is n o t a U2-grou p. F o r let

= G*H.

Then F = ? / [ & [G, tt]].

N o w a c o m m u t a t o r in G a l w a y s c o m m u t e s w i t h e v e r y e l e m e n t h E H : [ x - l y - l x y , h] = Ix, hi -1 [y, h] -~ Ix, h] [y, hi = 1 (x, y E G).

So [c, d] 2 = [c ~, d] = 1,

a l t h o u g h [c, d] =~ 1.

T H ~ O a ~ M 17.4. The second nilpotent product F o/E~-groups ]r (~ EA) is an E~-group.

Proo/. W e h a v e o n l y t o p r o v e t h e e x i s t e n c e of p t h r o o t s in ~ for e v e r y p E co, L e t t h e n g E F a n d let

g=g~(1)g~(2) ... g~(=)u, ~ ( 1 ) < ~ ( 2 ) < . . . < ~ ( m ) , uE[F~].

N o w e v e r y simple c r o s s - c o m m u t a t o r has a p t h root; for [/~ =, I,~F = [ ( h =)', I,,] = [ h , / , 8

since [F~.] ~ ~ (F). C o n s e q u e n t l y [F~] is itself a n E ~ - g r o u p . Choose u 1 E [F~] so t h a t u i ~ = u ' [ ( I-I

[g~(j>=,g~(~>z]~Pc~-')] ~.

T h e n i t follows t h a t

g 0 ~ ~ ( 1 ) 7~ 9 g~(2) ~ " ' " ~ l ( m ) ~ " U 1

is a p t h r o o t of g. This c o m p l e t e s t h e p r o o f of t h e t h e o r e m .

1 6 - 6 0 1 7 3 0 3 3 . A c t a mathematica. 104. I m p r i m 6 le 21 d 6 c e m b r e 1960

Références

Documents relatifs

For these experienced regional nationalists, the political purpose is to use the European argument to justify further concessions of policy competences that are

The kernel of the AMS is its library of Detailed Basic Models (DBMs). Each DBM was designed in order to comply with the most important criterion which is reusability. Indeed, each

Actually, an interface can be used twice even if the instance is connected to two different DBM instances; in this case, these two different DBMs belong to the same superfacet..

If the file does not contain errors, neither syntactic nor semantic, the ADL compiler generates two files: a “___.qnp” file which contains the QNAP2 model associated with the

the mean energy lost by charged particles in causing ionizations and excitations inside dm, This stems from the mechanisms in which ionizing particles – charged or

(2010) as an intermediate step to treat clarified water, while Treguer et al. These studies gave encouraging results regarding the potential of HMPs using aged PAC to remove NOM

piston compressors Chapter III : Lubricants used in oil injected screw air compressors Chapter IV : Solubility’s of gases and water in lubricants. Chapter V : Destructive processes

(1994-b), Fracture and crack growth by element free Galerkin methods, Modelling and Simulation in Material Science and Engineering, Vol.. BELYTCHKO T., KRONGAUZ Y., ORGAN D.,