34. T h e w o r k d o n e h i t h e r t o in C h a p t e r s V I a n d V I I is t h e basis for a n i n v e s t i g a t i o n of t h e p r o p e r t i e s of Do-free g r o u p s a n d Do-free p r o d u c t s of D o - g r o u p s .
The e x i s t e n c e of a Do-free g r o u p of a r b i t r a r y ~o-rank is e n s u r e d b y t h e r e s u l t s of B i r k h o f f [5] on a b s t r a c t a l g e b r a i c systems. T h e r a t h e r h a z y f o r m t h a t a Do-free g r o u p t a k e s on f r o m t h e e x i s t e n c e p r o o f is r e p l a c e d here b y a m o r e concrete r e a l i s a t i o n . W e p r o v e t h e i m p o r t a n t r e s u l t t h a t a free o - c l o s u r e of a free g r o u p of r a n k m is a D~o-free g r o u p of ~ - r a n k m. N o w a l t h o u g h a free o~-closure of a g r o u p (in t h e class No) b r i s t l e s w i t h t r a n s f i n i t e ordinals, we h a v e here a t h e o r e t i c a l l y u s a b l e r e a l i s a t i o n of a Do-free group.
W e m a k e use of this r e a l i s a t i o n t o p r o v e a n u m b e r of results a b o u t Do-free groups. I n p a r t i c u l a r we p r o v e t h a t a Do-free g r o u p is t o r s i o n - f r e e a n d t h e n , still more, we p r o v e t h a t a Do-free g r o u p is a n R - g r o u p , i n d e p e n d e n t l y of (o being t h e set of all p r i m e s or otherwise. A n i n t e r e s t i n g r e s u l t is t h a t t h e e e n t r a l i s e r of e v e r y n o n - t r i v i a l e l e m e n t of a Do-free g r o u p is i s o m o r p h i c to Fo. This e n a b l e s us t o d e d u c e t h a t a D o - g r o u p , w h i c h is n o t abelian, has t r i v i a l centre. W e p r o v e also t h a t a free ~0-generating set of a Do-free g r o u p g e n e r a t e s (in t h e u s u a l sense) a free group. This suggests t h e p o s s i b i l i t y t h a t a Do-free g r o u p is l o c a l l y free. H o w e v e r , i t t u r n s o u t t h a t t h i s is t r u e o n l y for Do-free g r o u p s w h i c h are abelian. W e p r o v e n e x t , b y a t r a n s f i n i t e i n d u c t i o n , t h a t t h e d e r i v e d g r o u p of a Do-free g r o u p is a n o - s u b g r o u p if, a n d o n l y if, i t is t r i v i a l . This p r o v i d e s t h e clue to t h e s t r u c t u r e of t h e f a c t o r g r o u p of a Do-free g r o u p b y its c o m m u t a t o r s u b g r o u p , w h i c h we t h e n d e t e r m i n e .
2 8 0 G I L B E R T B A U M S L A G
The existence of a D~-free product of D~-groups is taken care of b y a theorem of Sikorski [34] on general algebras. I t is, however, still an interesting result in its own right that the free co-closure of a free product of D~-groups is in fact a D~-free product of these groups; this statement implies simultaneously a proof of the existence of a D~-free product and a realisation of such a product as the union of an ascending sequence 0f U~-groups.
I n the same way as with D~-free groups we can utilise this realisation of a D~-free product of D~-groups to find some of their properties. I n particular, it turns out t h a t a D~-free product of groups isomorphic to P~ is a D~-free group (which is as it "should" be). We prove some simple properties of D~-free products. One interesting "carry over" from the realm of free groups is an analogous theorem to t h a t of Baer & Levi [2] for free products, namely: A D~-group cannot be decomposed in a non-trivial way simultaneously into a direct product of D~-groups and a D~-free product of D~-groups.
35. I n this section we shall prove, by means of elementary cancellation arguments only, that a free product of groups which belong to the class ~ is also in ~ . The proof follows a similar pattern to the proof of Theorem 29.1; it is, however, very much simpler. We shall effect the proof of the theorem by proving a number of lemmas and then make use of them to deduce this theorem.
We take now F to be the free product of groups F~, where 2 ranges over an index set A.
LEMMA 35.1. Let a E F be a cyclically reduced element o/ length at least two. Then i/
bE F (b:~ 1) commutes with a, it is also cyclically reduced o/length at least two.
Proo/. Let the normal forms of a and b be
~ ( 1 ) ~ ( 2 ) ~ ( m ) / a = % t ( 1 ) % ~ ( 2 ) 9 9 9 ~ 2 ( m )
p ( n ) .
/
(35.11)
Now a is cyclically reduced of length at least two and so 2 (1):~ )~ (m). We have
b - t $ ~ ~ p ( 1 ) t ' p ( 2 ) m ) ~(2) 9 9 9 ~(n) C(1) C(2) p ( n ) Jl(1) ) . ( 2 ) 9 9 9 ( ~ ) . ( m ) - - t~).(l) (~)~(2) 9 9 9 (~)l(m) ( ~ p ( 1 ) v g ( 2 ) .(m) __.(1) .(2) ~(m) t(:) ~(2) 9 - . d (n) p ( n ) - - - a b . (35.12)
If/~ (n) ~: 2 (1) then ~ (1) :t: ~ (m) and (35.12) yields
#(1) =;t(1) ~t(m)=#(n),
and so /~(1) =~ #(n);
S O M E A S P E C T S O F G R O U P S W I T H U N I Q U E R O O T S 281 therefore b is cyclically reduced of length at least two. On the other hand, if # (n) = ~ (1) then # (1) = ~ (m) and so/~ (1) # # (n) and b is again cyclically reduced of length at least two.
This completes the proof of the lemma.
LEM~A 35.2. L e t a E F be cyclically reduced, w i t h Z ( a ) = m >~ 2. Let b E C ( a , /) a n d suppose ;k (b) = n >~ m. P u t n = a m § fl (0 <~ fl < m). T h e n
a n d b can be written i n t h e / o r m
~-OorZ>~2,
where X (a*) = fl;
hence a* is either trivial or cyclically reduced o] length at least two.
Proo/. L e t a and b have normal forms given b y (35.11). I t follows from the proof of L e m m a 35.1 t h a t either ju(1)=~(1) and # ( n ) = 2 ( m ) or # ( 1 ) - ~ ( m ) and # ( n ) = ~ ( 1 ) . L e t us suppose # (1) = ~ (1) and # (n) = ~ (m). Then it follows immediately from (35.12) t h a t
/ ~ ( 1 ) = 2 ( 1 ) , /~(2)=2(2) . . . . , # ( m ) = ~ ( m )
~(1) ~ C(1), ~(2) ~ 0(2), . . . ~ ~(m) ~ c(m);
thus b = a b,
where X ( b ) = ( ~ - 1)m + f i and b is cyclically reduced of length at least two or trivial.
Furthermore, ~ commutes with a a n d so the process can be continued until b = a~a *,
fl =X(a*) <),(a). We assumed at the outset t h a t #(1) =~(1); if however ,u(1) = 2 ( m ) then we would have obtained
b = a-=a*;
these are the only two possibilities and so the l e m m a follows.
LEMMA 35.3. L e t a E F be cyclically reduced o / l e n g t h at least two. Let b E C (a, F ) (b # 1).
T h e n a a n d b are powers o / a c o m m o n element c E F .
Proo]. We suppose ~ (b) >/Z (a). Then b y L e m m a 35.2 we can write
b : a =~ a l ,
2 ~ 2 G I L B E R T B A U M S L A G
~k(al) <~,(a). N o w aEC(al) a n d ~(al) <~(a); so w e can apply the s a m e l e m m a to a I a n d a:
a = a~ ~ a2,
w i t h ~.(a2) < ~ (al). C o n t i n u i n g t h e process we o b t a i n i n t h i s w a y a set of e l e m e n t s a~ E F such t h a t
~.(a) > ~ ( a l ) > . - - > ~ . ( a i ) > . . . .
T h e process m u s t t e r m i n a t e w i t h ~ (a~) = 0 for some i, s a y i = j + 2. T h e n aj = a ~ 2 : c ~j+2, say.
I t follows t h a t b o t h a a n d b c a n be expressed as powers of c a n d this completes t h e proof of t h e l e m m a .
LEMMA 35.4. L e t a E F be cyclically reduced o/ length at least two. T h e n C(a) is a n i n / i n i t e cyclic group.
Proo/. Choose c E F so t h a t a = c T, with r as large as possible. Suppose b E C (a); t h e n , b y L e m m a 35.3,
a : d s, b = d t, where s a n d t are integers a n d d E F . T h e n
d s = c r. (35.41)
Now, b y t h e choice of r, s <~ r a n d so, on c o m p a r i n g t h e two sides of (35.41) we see t h a t t h e (cyclically reduced) e l e m e n t d has t h e form d = cc 1 (cl E F ) , where ;k (c) + X (Cl) = ~ (d).
H e n c e
(COl) s = c r = (clc) s.
T a k i n g the e x t r e m e r i g h t - h a n d - s i d e a n d e x t r e m e left-hand-side of this e q u a t i o n we h a v e
5 C 1 = C l C ~
a n d hence d = cc 1 c o m m u t e s w i t h c. R e m e m b e r i n g how c was chosen it follows from L e m m a 35.3 t h a t d - c u, for some i n t e g e r u. I t follows t h a t C (a) is t h e i n f i n i t e cyclic group genera- t e d b y c.
LEMMA 35.5. L e t p e w a n d let m a n d n be integers. S u p p o s e F is the /ree product o]
groups F~., each o / w h i c h belongs to the class ~ . S u p p o s e a E F does not have a p-th root and that
b - l a m b = a ~, (35.51)
/or some b E F . T h e n m = n.
S O M E A S P E C T S O F G R O U P S W I T H U N I Q U E R O O T S 2 8 3
Proo/. W e m a y a s s u m e a is c y c l i c a l l y r e d u c e d . L e t us r e c a s t (35.51) in t h e f o r m
arab = baL (35.52)
I f X(a) = 1 t h e n i t follows f r o m (35.52) t h a t X(b) = 1, a n d since e a c h f a c t o r F~ belongs t o
~ i t follows t h a t m = n. I f X (a) > 1 t h e n i t follows t h a t b m u s t be c y c l i c a l l y r e d u c e d a n d so we can a r r a n g e i t t h a t e i t h e r
X(amb) = X ( a m) + X ( b ) a n d X ( b a n) - X ( b ) § n) or X(a~b -1) = X ( a n) §
-I)
a n d X(b 1am) = X ( b -1) + X(am).Since a"b = ba n, if t h e first of t h e a b o v e s i t u a t i o n s occurs, we h a v e X(amb) = m X ( a ) + X ( b ) - n X ( a ) + X ( b ) = X ( b a ~)
a n d so m = n. A s i m i l a r a r g u m e n t h o l d s for t h e s e c o n d case a n d t h e l e m m a t h e n follows.
T H E O R E M 35.6. The /ree product F o/ groups F~, each o/ which belongs to the class
~ (~ CA) belongs also to the class ~ .
Proo/. W e k n o w a l r e a d y f r o m T h e o r e m 17.2 t h a t a free p r o d u c t of U~o-groups is a Uo~-group; t h e r e f o r e F satisfies 27.1.
S u p p o s e n e x t t h a t p Ew a n d t h a t a C F does n o t h a v e a p t h root. W e m a y s u p p o s e t h a t a is c y c l i c a l l y r e d u c e d . I f a lies in one of t h e f a c t o r s F~ t h e n
C (a, F) = C (a, F~)
a n d so C (a, F ) is i s o m o r p h i c t o a s u b g r o u p of Fo. On t h e o t h e r h a n d , if a is of l e n g t h t w o or m o r e t h e n , b y L e m m a 35.4, C(a, F ) is a n i n f i n i t e cyclic g r o u p . H e n c e F satisfies 27.2.
N o w let a be as a b o v e a n d let m be a n o n - z e r o integer. T h e n if a C Fx, C (a, ~ ) = C (a, ~'~) - C (a m, F~.) - C (a m, F ) .
I f a is c y c l i c a l l y r e d u c e d of l e n g t h t w o or m o r e t h e n a m is likewise c y c l i c a l l y r e d u c e d of l e n g t h a t ]east two. I t follows, b e c a u s e C (a) a n d C (a m) are infinite cyclic groups, a n d hence a b e l i a n (cf. L e m m a 35.4), t h a t
C (a, F) = C
(a m, F).
So F satisfies 27.3.
F i n a l l y , b y L e m m a 35.5, F satisfies 27.4 a n d 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 .
284 G I L B E R T B A U M S L A G
COROLLARY 35.7. Every/ree group belongs to 0~"
Proo/. An infinite cyclic group belongs to 0~; hence, b y Theorem 35.6, so does every free group.
We remark that had use been made of a theorem b y Kurosh [22] on the subgroups of a free product, the proof of Theorem 35.6 could have been achieved more easily. The proof we have given here will be used in 38 to provide an elementary proof of the theorem due to Baer & Levi [2] which we cited earlier.
36. We shall prove in this section some of the simpler properties of D~-free groups.
THEOREM 36.1. A /ree to-closure F* o/ a /ree group F o/ rank m is a D~o-/ree group o/ to-rank m. Moreover, every set o/ /ree generators o / F is also a set o//ree to-generators o/ F*.
Proo/. Let F be a free group of rank m and let X be a free generating set of F. Now b y Corollary 35.7 F belongs to ~ , and so its free to-closure exists and belongs also to
~ . Let now H be a D~-group and let 0 be a mapping of X into H. This mapping 0 can be extended to a homomorphism ~ of F into H, since X is a free generating set of F.
Theorem 33.4 can now be applied to extend ~ to a homomorphism T* of F* into H.
Now X generates F and F to-generates F* (Lemma 33.5); hence X to-generates F*.
X is in fact a free to-generating set of F*. Because, as we have seen above, for every D~
group H and every mapping 0 of X into H there exists a homomorphism ~* of F* into H which coincides with 0 on X. Therefore F* is a D~-free group freely to-generated by the set X.
Now a free to-closure of a group of order m is again of order m if m is infinite (cf.
Theorem 31.4 and the constructive process t h a t precedes it). I t follows from this observation and Theorem 36.1 that
TH]~OREM 36.2. A D~-]ree group F* ]reely to-generated by the non-empty set X is countably in/inite i/ X is/inite. I / X is in/inite then the order o / F * is equal to the number o/
elements o / X .
We shall make further use of Theorem 36.1 to prove a number of results concerning D~-free groups. But first we prove a sort of converse to it.
T H E O R E ~ 36.3. Let G* be a Do~-/ree group/reely to-generated by the set Y. Then the group G generated by Y is a/ree group/reely generated by Y.
Proo/. Let F be a free group freely generated b y a set X of the same cardinality as Y.
Further, let F* be a free to-closure of F; F* is then, b y Theorem 36.1, a D~-free group
SOME ASPECTS OF G ~ O U P S W I T t t U N I Q U E I~OOTS 2 8 5
freely w-generated b y the set X. Now there is a one-to-one mapping 0 of X onto Y; this can be extended to an isomorphism q0* of F* onto G* (cf. the proof of Theorem 25.2).
Now F maps under ~0" onto the group generated by Y i.e. onto G; hence G is free since F is free. Furthermore, X maps onto Y under ~* and since X is a free generating set of F,
Y is a free generating set of G. This completes the proof of the theorem.
We note at this point that it follows from the proof of Theorem 36.3 t h a t i/ G* is a D~-/ree group/reely o>generated by a set Y, then G* may be thought o / a s the/ree ~o-closure o/ the /ree group G/reely generated by Y.
Theorem 36.3 brings to mind the question as to whether D~-free groups are locally free. However, this is true only in the case of a D~-free group of ~o-rank one. We make use of the following lemma, which is interesting in its own right.
LEM~[A 36.4. Let n be an integer greater than one. Then a simple commutator in a / t e e group is an n-th power i/, and only i/, it is the identity.
Proo/. Let F be a non-abelian free group and suppose there is a non-trivial element / C F such that
/~ = [g, h i ,
g, h E F . Now F is locally infinite and so ] ~ 1; hence g and h do not commute.
Consider now the subgroup G of F generated b y / , g, h:
G = gp (I, g, h).
The Nielson-Schreier theorem for free groups states t h a t the subgroups of a free group are free (cf. e.g. Schreier [33]); so G is itself free. Now the factor group of G by its commu- tator subgroup G' is a free abelian group (cf. e.g. Kurosh [21]) of rank m, where m is the rank of G. Now/~ 6 G' and consequently, by the remark above, / C O'. Hence G/G" is of rank two, and is generated modulo G' b y g and h. Consequently G is itself of rank two. Therefore we can find two elements g* and h* such that they generate G and such t h a t g*G' = gG' and h*G' = hG'. Thus
g * = g g ' , h * = h h ' , g',h'EG'. (36.41) A theorem of Magnus [24] states t h a t if a free group of finite rank m is generated by m elements, then these m elements are in fact a free generating set; thus g* and h* are a free generating set of G.
Let now H be a nilpotent group of class two defined in the following way:
H = g p ( a , b ; a ~ = b ~ ' = l , [ a , b ] = a ~=b~).
2 8 6 GILBERT BAUMSLAG
I t follows f r o m its defining r e l a t i o n s t h a t t h e c e n t r e of H coincides w i t h its d e r i v e d g r o u p a n d is of o r d e r n.
L e t ~ be t h e h o m o m o r p h i s m of G i n t o H d e f i n e d b y g*~v = a, h * ~ = b.
I t follows t h e n f r o m t h e e q u a t i o n s (36.41) a n d t h e f a c t t h a t t h e d e r i v e d g r o u p of H lies in t h e c e n t r e of H t h a t [g, h ] ~ = [a, b]; for
[g, h]q~ = [g,g,-1, h , h,-1]c? - [ g , ~ g , - l % h,cph,-l~] = [g,% h*~] = [a, b].
N o w / n = [g, h] a n d so we h a v e shown t h a t
/n~ = [a, b ] # l ;
t h u s ]~p is of o r d e r n. H o w e v e r , ] lies in t h e d e r i v e d g r o u p of G a n d c o n s e q u e n t l y its i m a g e u n d e r ~ lies in t h e d e r i v e d g r o u p of H . B u t H' is of o r d e r n a n d t h e r e f o r e
/ ~ = (/~)~ = 1.
T h u s /n~ is s i m u l t a n e o u s l y a n e l e m e n t of o r d e r n a n d a n e l e m e n t of o r d e r 1, w h i c h is i m p o s s i b l e a n d so /~ c a n n o t be a c o m m u t a t o r . This c o m p l e t e s t h e proof of t h e l e m m a .
T H E O R E M 36.5. A Do,-/ree group is locally/ree i/, and only i/, it is abelian.
Proo/. A D~-free g r o u p w h i c h is a b e l i a n is i s o m o r p h i c t o F~ a n d so a n y f i n i t e l y gene- r a t e d s u b g r o u p is n e c e s s a r i l y cyclic, a n d hence free. T h u s a D~-free g r o u p w h i c h is a b e l i a n is l o c a l l y free.
On t h e o t h e r h a n d , l e t G* be a D~-free g r o u p w h i c h is n o t a b e l i a n . T h e n t h e r e e x i s t g, bEG* such t h a t [g, h] 4 = 1. L e t p e a ) a n d let / d e n o t e t h e p t h r o o t of [g, hi:
/~
= [g, h i .T h e n H = gp (/, g, h) is n o t free, b y L e m m a 36.4, a n d so G* is n o t l o c a l l y free.
W e p r o v e n e x t t h e following t h e o r e m .
T H E O R n M 36.6. A D~-/ree group is locally infinite.
Proo/. A free g r o u p is l o c a l l y infinite a n d hence, b y L e m m a 33.7, so is its free o)-closure.
T h u s a D~-free g r o u p is l o c a l l y infinite.
T ~ E o R ~ I 36.7. The centraliser o/ every element di//erent /rom 1 in any D~-/ree group is isomorphic to F~.
SOME ASPECTS OF G R O U P S ~VITIt U N I Q U E ROOTS 2 8 7
Proo/. T h e c e n t r a l i s e r of e v e r y e l e m e n t d i f f e r e n t f r o m 1 in a free g r o u p F is a n infinite cyclic g r o u p - - t h i s follows e a s i l y b y e.g. L e m m a 3 5 . 4 - - w h i c h is i s o m o r p h i c t o a s u b g r o u p of F~. H e n c e t h e c e n t r a l i s e r of e v e r y n o n - t r i v i a l e l e m e n t in its free co-closure is i s o m o r p h i c t o F~ ( L e m m a 33.9); 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 36.8. The centre o/ a D~-/ree group which is not abelian is trivial.
Proo/. L e t F * be a n o n - a b e l i a n D~-frce group; t h e n t h e co-rank m of F * is g r e a t e r t h a n 1. L e t X be ~ free ~ - g e n e r a t i n g set of F * a n d choose xl, x 2 to b e d i s t i n c t e l e m e n t s of X . T h e n C (xl) a n d C (x~) are, b y T h e o r e m 36.7, i s o m o r p h i c to Fo, a n d hence
cl~ (xl) - C (xl) a n d cl~ (x2) = C (x2). (36.81) B u t X is ( o - i n d e p e n d e n t (see t h e e n d of 25) a n d t h e r e f o r e
cl~ (xl) n cl~ (x2) - 1.
I t follows f r o m (36.81) t h a t
C (xl) N C (x2) - 1. (36.82)
N o w t h e c e n t r a l i s e r of a n y e l e m e n t in a g r o u p c o n t a i n s t h e c e n t r e of t h a t group; conse- q u e n t l y so does t h e i n t e r s e c t i o n of t h e c e n t r a l i s e r s of a n a r b i t r a r y n u m b e r of elements. I n p a r t i c u l a r
~ ( F * ) ~< C(xl) n C(x2);
so b y (36.82) t h e c e n t r e of F * is t r i v i a l .
T H E O R E M 36.9. The normaliser o / t h e centraliser o / a n y element, di//erent /tom 1, in a D,~-/ree group F* coincides with the centraliser, and is there/ore isomorphic to F~.
Proo/. L e t 1 -~ a C F * a n d s u p p o s e y C F * n o r m a l i s e s C (a). Since C (a) is i s o m o r p h i c t o F o (Theorem 36.7), i t is l o c a l l y cyclic. T h u s
a a n d y - l a y a r e p o w e r s of a c o m m o n e l e m e n t b:
T h u s
a - b 'n, y - l a y - b n.
y-lb'ny - bL
Since F * C ~o~ it follows t h a t m = n a n d so y E C (a). This c o m p l e t e s t h e proof of t h e t h e o r e m .
2 8 8 GILBERT BAUMSLAG
Finally we generalise Theorem 36.6; two simple proofs present themselves and so we give them both.
THEO~]~M 36.10. Let ~o be any non-empty set o/primes. Then every D~,-/ree group is an R-group.
Proo/. (i) A free group is an R-group (Theorem 17.2). Furthermore, a free w-closure of an R-group is an R-group (Lemma 33.8). The theorem now follows immediately on applying Theorem 36.1.
(ii) Let F* be a D~-free group, l e t / , g E F*, and let n be a positive integer. Suppose t h a t
/n = gn.
N o w / E C ( g ~) = C(g) (by Theorem 36.7). Hence (/g-l) n : 1;
but, by Theorem 36.6, F* is locally infinite and s o / g -1 = 1. Therefore / = g and this com- pletes the proof of the theorem.
37. Theorem 25.1 states that the factor group of a D~-free group of w-rank m b y the w-closure o / i t s commutator subgroup is a direct product of m isomorphic copies of l ~ . I n this section we shall completely determine the structure of the factor group of a D~- free group by its commutator subgroup. We shall need the notion of a restricted direct product of groups with an amalgamated subgroup; this notion was introduced by B. H.
Neumann and Hanna Neumann [30]. Only a particular case of this product is needed for our purpose; it is this case which we define here. Suppose A and B are given groups and H ~< $(A), K ~< ~(B). Suppose further t h a t K ~ H =~L, where L is some given group. Let 0 be an isomorphism of H to K. P u t M = A x B and put
N = g p ( m = a b -1 ; m E M , a E H , b e K , aO =b).
Then N is normal in M and so we can form the factor group D = M / N ; D is called the direct product o I A and B with L amalgamated, or the generalised direct product o/ A and B (the amalgamation being understood); we shall write
D = { A x B ; L } .
D is generated by isomorphic copies of A and B which intersect in a group isomorphic to L. When dealing with such a product we shall usually identify groups with their isomorphic
copies (as is usually done in the case of a direct product of groups).
SOME ASPECTS OF GROUPS WITH UNIQUE I%OOTS 2 8 9 LEMMA 37.1. Let F be a non-abelian /roe group and let c be an clement o/ F which is not a p-th power, /or some p in r Let, /urther, c be a member o/the second term o/the lower central series o] F but not a member o] the third term o/the lower central series o] F. Let P be a supcrgroup o] the (necessarily cyclic) ccntraliscr o / c in F such that P intersects F in C (c, F) and such that there is an element c o in P satis/ying c~ = c. Finally let G be the generalised /roe product o/ F and P. Then the p-th root c o o / c does not lie in the commutator subgroup o / G . Proo/. L e t 0 denote the n a t u r a l h o m o m o r p h i s m of F onto F/(2)F, where (*)F here denotes the (i + 1)st m e m b e r of the lower central series of the free group F . N o w (1)F/(e)F is a direct p r o d u c t of infinite cyclic groups (by a theorem of W i t t [37]) and, since c ~(2)F, [C (c, F)] 0 is an isomorphic c o p y of t h e cyclic centraliser C (c, F). Take P + to be a super- group of C (c, F ) 0 which is isomorphic to P in such a w a y t h a t the isomorphism between t h e m m a p s C(e, F), qua subgroup of P, onto [C(c, F)]O in the same w a y as 0 m a p s C(c, F ) onto [C (c, F)] 0; let, further, P ~ have intersection [C (c, F)]O with FO.
N o w C(c, F)O <~ ~(FO) a n d so we can form the generalised direct p r o d u c t D of FO a n d P+:
D = ( F 0 ~P+; C(c, F)0}.
B y definition of the generalised free p r o d u c t it follows t h a t we can e x t e n d the homo- morphisms 0 (of F onto FO) a n d ~ (of P onto P+) to a h o m o m o r p h i s m F of G into D. N o w if c o ~ G', t h e n c0F C D'. However, c0F # 1; furthermore,
D ' = ( F ~ ) ' ,
since P + is abelian. B u t CoyJEP§ a n d Co~fl~FO a n d so, in particular, CoyJr in other w o r d s
CoyJ r D ' . Hence c o ~ G' a n d so we h a v e p r o v e d the lemma.
T ~ . o n ] ~ M 37.2.(1) The commutator subgroup o / a D~-/ree group is an w-subgroup i/
and only i/, it is trivial.
Proo/. L e t F * be a D~-free group; we shall t a k e F * to be a free ~o-closure of a free group F . I t is clear t h a t the trivial subgroup is an ~o-subgroup; so when the c o m m u t a t o r subgroup K of F* is trivial there is n o t h i n g to prove.
(1) Thus we have examples of D~-groups whose derived groups are not o)-subgroups (see 14).
2 9 0 G I L B E R T BAUMSLAG
Suppose, on t h e o t h e r h a n d , t h a t K is n o n - t r i v i a l . T h e n F * is a n o n - a b e l i a n
Do~-free
g r o u p . So if X is a free g e n e r a t i n g set of F , t h e n I X I > 1. W e m a k e use now of t h e f a c t t h a t F * is a free w-closure of F to w r i t e F * in t h e f o r m (using (33.3)):
F * = U F~.
~r
I n o r d e r t o p r o v e t h a t K is n o t a n o~-subgroup we h a v e t o p r o v e t h a t for some p e w t h e r e is a n e l e m e n t
coEF*
such t h a t c~ =ceK
b u tco~K.
W e choose a s i m p l e c o m m u t a t o rce(2)F,
w i t hc(~)F,;
t h e n , b y L e m m a 36.4, c is n o t a p t h p o w e r in t h e free g r o u p F , a n d so co, t h e p t h r o o t of c, does n o t lie in F , W e shall p r o v e t h a tco(~K
a l t h o u g h , b y our choice of Co, c~ = c e K .The o r d e r i n g in F can be chosen so t h a t t + ~ C .
T h e n F,~ = ( F , * P ; A}.
W e m a k e use of L e m m a 37.1 in a s s e r t i n g t h a t
coCF',+
since c0~_F[+. So we h a v e t h e first s t e p in a p r o o f b y t r a n s f i n i t e i n d u c t i o n . L e t us n o w s u p p o s e t h a t t + < f l e ~4 a n d t h a t c o r F : for all ~ < fl, ~ E A . I f fl does n o t h a v e a p r e d e c e s s o r t h e na n d since F~ = U F~
Co ~ F~. If/~ does h a v e a predecessor, say/~-, t h e n t h e r e a r e t w o possibilities: E i t h e r F z = 2'Z-, in w h i c h case c o r F~, or
= ; 4 §
W e shall p r o v e t h a t t h e r e is a h o m o m o r p h i c i m a g e of F~ in w h i c h t h e i m a g e of co is n o t in t h e c o m m u t a t o r s u b g r o u p , which shows t h a t c o is n o t in t h e c o m m u t a t o r s u b g r o u p of F z.
Consider t h e f a c t o r g r o u p
G =
N o w G is a n o n - t r i v i a l a b e l i a n g r o u p since c o ~ - ~ - . F u r t h e r m o r e , t h e p k t h r o o t s of co, CO, COY'g, C0~2~ ...~
where Co ~ d e n o t e s t h e p % h r o o t of c o, g e n e r a t e m o d u l o F ~ - a g r o u p H i s o m o r p h i c t o Z (/9~).
N o t e t h a t all t h e p k t h r o o t s of c o lie in F a - since t h e y lie a l r e a d y in F,*, a n d b y t h e choice
SOME ASPECTS OF GROUPS WITH UNIQUE I~OOTS 291 of /~, t + ~</7 . N o w H is a d i v i s i b l e s u b g r o u p of t h e a b e l i a n g r o u p G a n d c o n s e q u e n t l y i t s p l i t s off as a d i r e c t f a c t o r (cf. e.g. K a p l a n s k y [17] p. 8):
G - H •
I t follows t h a t FZ- can be h o m o m o r p h i c a l l y m a p p e d o n t o H b y a h o m o m o r p h i s m 0 so t h a t coO =4= 1.
Consider now A + : 0 induces a h o m o m o r p h i s m of A + i n t o H . T h e r e a r e t w o p o s s i b i l i t i e s t h a t can occur:
i) A+O = 1;
ii) A+O :t: 1.
I n i) i t is e a s y to see t h a t c o @ F~. F o r we can define here a h o m o m o r p h i s m ~ of P + i n t o H w h i c h coincides w i t h 0 on A + s i m p l y b y s t i p u l a t i n g t h a t all t h e m e m b e r s of P + m a p o n t o 1 u n d e r 9- T h e n b y t h e d e f i n i t i o n of t h e g e n e r a l i s e d free p r o d u c t we c a n e x t e n d 0 a n d s i m u l t a n e o u s l y to a h o m o m o r p h i s m ~ of F~ o n t o H . N o w Co~ f ~ e o O 4 1; b u t H ' = 1 a n d so e0r
I n ii) we h a v e for some a EA§ s a y %, %0 = 1. W e a d d t h i s r e l a t i o n t o t h e defining r e l a t i o n s of P+; t h i s y i e l d s a t o r s i o n g r o u p /5. This g r o u p /5 has a s u b g r o u p i s o m o r p h i c t o Z (p~) a n d so H is a h o m o m o r p h i c i m a g e of P+. T h u s we c a n f i n d a h o m o m o r p h i s m ~ of P + o n t o H w h i c h coincides w i t h 0 on A +. The d e f i n i t i o n of t h e genera]ised free p r o d u c t ensures t h a t we can e x t e n d , s i m u l t a n e o u s l y , t h e h o m o m o r p h i s m s 0 a n d qJ, r e s p e c t i v e l y , of F f - a n d P + i n t o H to a h o m o m o r p h i s m ~ of F~ i n t o H . :Now a g a i n c0~ ~ 1, a n d so, because H ' = 1, c 0 r
W e are t h e r e f o r e e n t i t l e d , w i t h t h e a i d of a t r a n s f i n i t e i n d u c t i o n , t o d e d u c e t h a t c o r F : for all ~ E A . H e n c e
Co r
U F:
= ( F * ) ' = K .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 .
T h e o r e m 37.2 e n a b l e s us t o p r o v e t h e following i m p o r t a n t result:
THEOlCEM 37.3. The /actor group o/ a non-abelian D~-/ree group F* o/~o-rank m by its commutator subgroup K splits into a direct product o/ a divisible torsion group Q and a torsion/tee group R:
F * / K = @ • R .
When m is/inite, the torsion group Q is a direct product o / a countably in/inite number o/
groups isomorphic to Z (p~) /or each p in co. When m is in/inite then Q is a direct product o/
1 9 - 60173033. A c t a mathematica. 104. I m p r i m ~ le 21 d 6 c e m b r e 1960
292 G I L B E R T B A U M S L A G
m groups isomorphic to Z ( p ~) /or each p in co. Finally the torsion-/ree group R is a direct product o / m groups isomorphic to F~.
Proo/. T h e g r o u p F * / K is a b e l i a n a n d , as such, i t can be s p l i t i n t o a d i r e c t p r o d u c t of a m a x i m a l d i v i s i b l e g r o u p A a n d a g r o u p B w h i c h c o n t a i n s no d i v i s i b l e s u b g r o u p s (cf.
e.g. K a p l a n s k y [17] p. 9):
F * / K = A • B.
L e t us s u p p o s e n o w t h a t F * is a free co-closure of t h e free g r o u p F a n d l e t X ' be a free g e n e r a t i n g set of F ' consisting of d i s t i n c t simple c o m m u t a t o r s of F (cf. L e v i [23]).
Choose cEX'; t h e n c is n o t a p t h p o w e r in F , b y L e m m a 36.4. I t follows f r o m t h e m e t h o d of p r o o f of T h e o r e m 37.2 t h a t t h e p k t h r o o t s of c:
C2/:~ C:7/:2~ . . .
d o n o t b e l o n g t o K a n d so t h e y g e n e r a t e m o d u l o K a g r o u p i s o m o r p h i c t o Z(p~). This a r g u m e n t holds for e v e r y p e w ; so for e a c h p e w t h e p % h r o o t s of c g e n e r a t e m o d u l o K a Z (pOe) a n d hence, l e t t i n g p r u n t h r o u g h t h e whole of co, t h e p % h r o o t s of c g e n e r a t e m o d u l o K a d i r e c t p r o d u c t of g r o u p s i s o m o r p h i c t o Z (p~). I f m is finite, t h e n F ' is of c o u n t a b l e r a n k (cf. L e v i [23]). T h u s X ' is c o u n t a b l y infinite a n d so we h a v e a c o u n t a b l y infinite n u m b e r of i n d e p e n d e n t choices for c a n d hence, in t h i s case, A c o n t a i n s a s u b g r o u p w h i c h is a d i r e c t p r o d u c t of a c o u n t a b l y infinite n u m b e r of g r o u p s i s o m o r p h i c to Z ( p ~162 for e a c h p in co.
Similar]y, in t h e case where m is infinite, A c o n t a i n s a s u b g r o u p w h i c h is a d i r e c t p r o d u c t of m g r o u p s i s o m o r p h i c t o Z ( p ~) for e a c h p e w . W e t a k e Q t o be t h a t s u b g r o u p of A g e n e r a t e d b y all t h o s e s u b g r o u p s of A w h i c h are i s o m o r p h i c t o a Z ( p ~) for some p e a ) ; t h e n Q is a d i r e c t p r o d u c t of t h e s e s u b g r o u p s i s o m o r p h i c t o Z ( p r162 (cf. e.g. K a p l a n s k y [17] p. 8). W e can n o w s p l i t A i n t o a d i r e c t p r o d u c t of its d i v i s i b l e s u b g r o u p Q a n d a c o m p l e m e n t a r y f a c t o r C. N o w a d i r e c t f a c t o r of a d i v i s i b l e g r o u p is itself d i v i s i b l e (cf.
K a p l a n s k y [17]) a n d so C is divisible. T h u s
F * / K = Q z C • B.
N o w C • B is a g r o u p in w h i c h no e l e m e n t has o r d e r p, where p is a n y p r i m e in oJ.
F o r if / E F* g e n e r a t e s m o d u l o K a s u b g r o u p of o r d e r p in C • B t h e n its p % h r o o t s g e n e r a t e a s u b g r o u p of C • B i s o m o r p h i c t o Z ( p ~) a n d so in f a c t / K E Q . T h u s C x B is a n a b e l i a n g r o u p w i t h o u t e l e m e n t s of o r d e r p for all p Eco. C o n s e q u e n t l y i t is a Uo-group. T h e r e f o r e (Q • C • B ) / Q ~- C x B is a Uo-group a n d t h u s
F * / c l o (K) ~ C • B.
SOME ASI'ECTS OF GICOUPS WITH 17NIQITE ROOTS 293 Theorem 25.1 then informs us t h a t C • B is isomorphic to a direct product of m groups isomorphic to Fo, We p u t now R = C • B.
Thus we have, in both cases, a decomposition F * / K = Q • R,
where Q and R are of the required form. This completes the proof of the theorem.
38. We begin this section b y recasting some of the concepts connected with generalised free products (see 16) into analogous concepts for Do~-groups.
L e t G* be a D~-group and let G~ be ~-subgroups of G*, where ~ ranges over an index set A; suppose t h a t ~ = U G~ w-generates G* (note t h a t each G~, is itself a D~-group).
~A
Then we call G* the generalised D~-free product of its w-subgroups Ga (or, more simply, the generalised D~-free product of the Ga) if for every D~-group W a n d every set of homo- morphic mappings Fa of each Ga into W, every two ?a, ~ of which agree where both are defined, there exists a homomorphic m a p p i n g ~* of G* into W t h a t coincides with ?z on each G;. Now suppose G* is the generalised D~-free p r o d u c t of its w-subgroups G~ (2EA) and p u t
a~ n a , = H ~ , ( :
H,~).
where 2, # EA (2 4:/2). If all the intersections H ~ coincide to form a single subgroup H:
G ~ N G , = H ,
then G* is called the (generalised) D~-free product of the G~ with an amalgamated subgroup H; note t h a t H is itself a D~-group. I n the ease where H = 1, the trivial group, G* is called simply the D~-/ree product or, to emphasise the distinction, the ordinary Do,-free product of the G~.
Following B. H. N e u m a n n [27], who proves a like result for the generalised free product of groups, we can prove the "uniqueness" of the generalised D~-free product.
The proof of our theorem is similar to the proof of B. H. N e u m a n n ' s theorem and is therefore omitted.
THEOa],~V~ 38.1. Let G* be the gene~ulised D~-]ree product o~ its w-subg~vups G~ (~EA).
Let H* be the generalised Do~-/ree product o/ its w-subgroups HA (2EA). Then, i] ]or each
~eA, there is an isomorphism q~ o] G~ onto H~, every two 9~, ~)~ o/ which agree where both are de]ined, then all the q)~ can be extended simultaneously to an isomorphism o/ G* onto H*.
L e t now D~,~-groups G~ be given, where ~ runs over a suitable n o n - e m p t y index set A.
I n every G;. and to every index lu EA let an w-subgroup H ~ be distinguished; H~.z is always
294 G I L B E R T B A U M S L A G
t a k e n to be the whole group G~. I f there exists a group G* which is the generalised D~-free product of groups Gh with intersections
and if there are isomorphic mappings ~ of G~ onto G~,
such t h a t aLvays / ~ = H~, ~ ,
then we say t h a t the generalised D~-/ree product o] the Gz with amalgamated H~, (or s i m p l y the generalised D~-/ree product o / t h e G~) exists. I t is often convenient when dealing with generalised D~-free products to distinguish only between groups lying in different isomor- phism classes; we shall a d o p t this procedure whenever it is convenient and also not am- biguous.
The generalised free product of groups with a single subgroup a m a l g a m a t e d alway8 exists. However, it is not even true t h a t the generalised D~-free product of two D~-groups with a single co-subgroup a m a l g a m a t e d always exists.
For let A = gp (s, a, b, c ; R1), where
R 1 = { a = = b , b = = c , c s = a , s ~ = a 3 = b a = c 3 = [ a , b] = [ a , c ] = [b, c ] = 1 }
and let where
B = gp (t, a, b, c; R2),
R~ = {b t = a -1, a t = c -1, c t = b, t a : 1}.
I t can easily be verified t h a t both A and B are of order 81; hence, b y Corollary 11.6, t h e y are also b o t h a-groups. Now p u t
H = gp (a, b).
Then H is clearly an o~-subgroup of both A and B (here ~o = {2}).
Suppose, if possible, t h a t the generalised D~-free product F of A and B with H amal- gamated exists. Thus obviously
F t> gp(A, B).
I n particular both belong to F. Now
/ = s t a , g = s t
12 = s t a . s t a = s t s . b t a = s t s t . a - l a = s t s t = f .
SOME ASPECTS OF GROUPS W I T H U N I Q U E ROOTS 2 9 5
B u t /=# g and so F is not a a-group, a contradiction. Thus this example shows t h a t the generalised D~-free product of two /)~-groups does not
always
exist. However, in the particular case of the ordinary Doe-free product we can in fact prove t h a t this product always exists.C)THEOREM 38.2. Let G~ be given D~o-groups, where A ranges over an index .set A. Then the /tee co-closure G* o/the/tee product G o/the G~. is their ordinary D~-/ree product; hence
THEOREM 38.2. Let G~ be given D~o-groups, where A ranges over an index .set A. Then the /tee co-closure G* o/the/tee product G o/the G~. is their ordinary D~-/ree product; hence