The direct product of a hopfian group with a group with cyclic ccnter
R. I-Ims~o~
Polytechnic ]_nstitu~e of Brooklyn, Brooklyn, :N.Y., U.S.A.
Abstract
I n this p a p e r we c o n t i n u e o u r s t u d y of h o p f i c i t y b e g u n in [1], [2], [3], [4] a n d [5].
L e t A be h o p f i a n a n d let B h a v e a cyclic c e n t e r o f p r i m e p o w e r order. W e i m p r o v e T h e o r e m 4 of [2] b y showing t h a t if B has f i n i t e l y m a n y n o r m a l s u b g r o u p s which f o r m a chain (we s a y B is n - n o r m a l ) , t h e n A x B is h o p f i a n . W e t h e n consider t h e case w h e n B is a p - g r o u p o f n i l p o t e n c y class 2 a n d show t h a t in c e r t a i n cases A X B is h o p f i a n .
Some notations and preliminary results
W e d r a w f r e e l y f r o m t h e ideas in [1], [2] a n d [3]. I n t h e sequel ~ will designate a s u r j e c t i v e e n d o r m o r p h i s m o f A x B. I n o u r proofs we b e g i n b y a s s u m i n g t h a t A X B is n o t h o p f i a n so we m a y a s s u m e ~ is n o t a n i s o m o r p h i s m o n A. W e use t h e idea of t h e h o m o m o r p h i s m ~ , i n t r o d u c e d in [3]. W e n o t e t h a t for ~ , t o exist as i n t r o d u c e d in [3], we m u s t h a v e ~ a n i s o m o r p h i s m on B a n d B ~ f3 B = 1.
T h e s y m b o l s a, al, a 2 . . . will d e s i g n a t e e l e m e n t s in A.
LE~MA 1. Suppose Z ( B ) is cyclic of p r i m e power order pJ and Bod N B = 1, cr an isomorphism on B , j = 1, 2. Let <b> ~ Z ( B ) -~ Z and let (bta)~ = b, T h e n i f ~ is not an isomorphism on A, t = l m o d p .
_Proof. L e t ba = albr W e claim t h a t q ~ 0 m o d p . F o r o t h e r w i s e A B o ; - ~ A X <b> so t h a t Bc~ = <b~> x A 1"1 B~. B u t Bee has a cyclic c e n t e r a n d c a n n o t d e c o m p o s e n o n t r i v i a l l y as a d i r e c t p r o d u c t . T h u s B ~ B ~ ---- <b~> w h i c h c o n t r a d i c t s T h e o r e m 3 o f [1]. N o w
A = (A x Z ) ~ , = A a , < a l > . (1)
232 ~. HII~SttO.N ~
L e t cr 1 = (a~bf)~. H e n c e (a2bf-~)~ is a p : t h p o w e r ~nd so t h e r e f o r e is (a2bf-1)a,.
This i m p l i e s f - ~ 1 m o d p , for o t h e r w i s e if L = <a2bf-1 >
A ~- (A • = ( A L ) ~ , = A ~ , L ~ , . (2) B u t since L ~ , is g e n e r a t e d b y a p : t h p o w e r we m a y conclude f r o m (1) a n d (2) t h a t A - - - A ~ , w h i c h is impossible. N o w we s h o w t h a t ( t , p ) = 1. :For if M - = <bta> t h e n
( A M ) a , = ALx, M ~ , = A~x<al} = A . (3) B u t
A M = A • <b~> (4)
so t h a t i f pit, we could d e d u c e f r o m (3) a n d (4) t h a t A = A ~ , . Since ( t , p ) = 1, we see <b~> c Ac~B C Aa(bo~> so
AocB = A~<b~} := A~<b>. (5)
Also since (p, q) = p , we see t h a t B . A ~ = A~@h>. H e n c e we m a y w r i t e b = a z ~ . a ~ . B u t t h e n a3a , = a ~ - " so t h a t ( p , l - - r ) = p or else A = A a , . N o w l e t c = b'aa~lb -f. L e t P b e t h e s u b g r o u p of p : t h p o w e r s in Z ( A ) x Z ( B ) . N o t e
c~ 2 = aa~ 2 rood P . (6)
I f t ~ f m o d P, t h e n f r o m (6) we see t h a t A = ( A - <ca~1>)(cr = A(c~2),. I-~ence t = = - f ~ l .
Some results
TI~EORE~ l.
If
B is n-normal, A x B is hopfian.Proof. S u p p o s e t h e t h e o r e m is false. L e t C be as in [3], p. 182 a n d l e t w b e a g e n e r a t o r f o r Z(C). L e t ~ b e a n e n d o m o r p h i s m o f A • C w h i c h is n o t a n iso- m o r p h i s m on A. T h e n , as in (5) o f L e m m a 1, A ~ C = Aa<w>. B u t t h e n C ~--
<w>C f3 Ao~. I-Ience C c A ~ or C = <w>. T h e f o r m e r is i m p o s s i b l e b y L e m m a 4 of [2] a n d t h e l a t t e r is i m p o s s i b l e b y T h e o r e m 3 o f [1].
LE~MA 2. Let B be a group with generators b, xl, yj, x2, Y2 . . . . xn, y~ and defining relations of the f o r m :
[xl, y~] : b ph, x7 ~ : b nl, y~C~ = bm~, b~k : 1 , (7) and all other generator s commute. T h e n Z ( B ) : <b> and any automorphism of Z ( B ) may be extended to an automorphism of B.
T I I E D I R E C T P R O D U C T O F A H O P F I A N GI~OUI:' W I T H A. G R O U P W I T H C Y C L I C C E N T E R 233
_Pro@ This is a n u n p u b l i s h e d r e s u l t of F. Pickel. F o r t h e d u r a t i o n o f this p a p e r B will be as in t h e a b o v e L e m m a a n d p will be a p r i m e d i s t i n c t f r o m 2.
LEMMA 3. I f B is f i n i t e we m a y assume B Cl B ~ ~ = 1 f o r all n sufficiently large.
Proof. Otherwise as in L e m m a 5 of [1], p. 237, we can p r o d u c e a n o r m a l sub- g r o u p B 1 # 1 o f B w i t h Blc~ ~ ~ B 1 for some m > 0. H e n c e c~ "~ is a n i s o m o r p h i s m on B. B u t t h e n A N B ~ = 1 for BI~ ~ m u s t i n t e r s e c t a n y n o n t r i v i a l n o r m a l s u b g r o u p o f B ~ m n o n t r i v i a l l y . H e n c e A • A • A~'~B~ m a n d h e n c e A is isomorphic to a h o m o m o r p h i c i m a g e of Acr m.
THEOtgEM 2. I f j~ = r, i = 1, 2 , . . . n i n (7), then A • is hopfian.
P r o @ B y L e m m a 3 we m a y a s s u m e B C l B c J = 1, i = 1,2. L e t ( b t a ) ~ = b . F r o m L e m m a 3 we see t h a t we m a y a s s u m e t 2~ I m o d p, for otherwise we m a y replace ~ b y Oct w h e r e 0 is a s u r j e c t i v e e n d o m o r p h i s m of A • B w h i c h is t h e i d e n t i t y on A a n d which acts as a suitable i s o m o r p h i s m on B. B y L c m m a 1, is n o t a n i s o m o r p h i s m o n B. L~t K = (kernel ~) (3 B. T h e n cr i n d u c e s a sur- j e c t i v e e n d o m o r p h i s m Y o f A + B / K o n t o A + B / K in t h e n a t u r a l w a y . L e t P = { g P l g 6 B } . I f g 6 B , we set g ' = g K in B / K = B , a n d claim
Z ( B , ) ~- <b', g;, g'2, . . . , g'k>
for suitable g~ in P . T o see this, suppose w is a w o r d in t h e g e n e r a t o r s of B such t h a t w' 6 Z ( B / K ) . L e t s be t h e e x p o n e n t of Yl in w. T h e n [xl, w] = [xl, yl] s 6 K . I f ( p , s ) = 1 t h e n [x1,?/1]= b P r 6 K - H e n c e [ x ~ , y ~ ] 6 K for all i.
H e n c e B . is abelian c o n t r a r y to T h e o r e m 3 o f [1]. H e n c e s is divisible b y p.
Similarly, t h e e x p o n e n t o f a n y o t h e r x or y s y m b o l in w is divisible b y p. N o w let
b~ = b~a~. (8)
Since cr is n o t a n i s o m o r p h i s m on B, c~ is n o t a n i s o m o r p h i s m on Z ( B ) os t h a t (p, q ) = p. Moreover, in view of (8)
A = (A • < b ' > ) r , = A T , < a l > . (9) N o w (b'-~a)~x = b~-qax I. B u t t h e n we see, ((b')'-~a)7, = a l ~ a n d since t ~ 1 m o d p,
A = (A X <b'>)7, = [A<(b')'-~a>]7, = A T , < a q > . (10) Since (p, q ) = p, (9) a n d (10) i m p l y A = A ~ , .
TttEOlCE)~ 3. I n (7) let JI <~ J2 <-- . . . ~ j,,-1 < j , < k a n d e~ = k - j~. S u p p o s e f u r t h e r that m~ - - n~ ~ 0 m o d pJ~, i = 1, 2 . . . . n. T h e n A • B is hopfian.
234 ~. ~iRs~o~
P r o o f . Our n o t a t i o n is as in t h e p r e v i o u s t h e o r e m . I f b p J" ~ K, as in T h e o r e m 2,
! t t
we c o u l d d e d u c e t h a t Z ( B . ) --= <b', gl, g2 9 9 9 gk>, gi C Pj a n d we could t h e n p r o c e e d e x a c t l y as before. W e m a y a s s u m e b Pi" E K . L e t e be t h e least positive i n t e g e r such t h a t for t = p i , , b ' E K . L e t
F = <xl, 5'1, x2, Y2, 9 9 9 xo-1, Y~-I, b> ,
A 1 = <A, x,, y~, x~+l, y~+l, 9 9 9 x , , y,>/<b'>, B I = F / < b ' > .
T h e n t h e d e f i n i n g relations for B i m p l y ( A • 2 1 5 1. N o t e t h a t A 1 is t h e direct p r o d u c t o f A a n d a f i n i t e a b e l i a n g r o u p so t h a t A~ is h o p f i a n b y T h e o r e m 3 o f [1]. N o w a i n d u c e s a n e n d o m o r p h i s m 7 o f A I • 1 in t h e n a t u r a l w a y . I f K x is t h e kernel o f 7 fl 171, a n d B~ = B 1 / K 1, t h e n 7 induces a surjective e n d o r m o r p h i s m 0 o f A I • 2 in t h e n a t u r a l way. N o w let F 1 = kernel a f l _~
a n d K 1 = F1/<b~>. T h e n JB2 ~ F / F 1. W i t h t h e aid of t h e d e f i n i t i o n o f e we see t h a t t h e center o f l~]F 1 is g e n e r a t e d b y e l e m e n t s b F 1, g l F x . . , g , F 1 for c e r t a i n g~ E F w h e r e each g~ is a p : t h p o w e r in F . W e m a y n o w m i m i c k the p r o o f o f T h e o r e m 3 a n d o b t a i n AIO , = A 1 a c o n t r a d i c t i o n o f t h e h o p f i c i t y o f A 1.
References
1. J=~IttSHOlg, R., Some theorems on hopfieity. Trans. Amer. Math. Soc., 141 (1969), 229--244.
2. --~)-- On hopfian Groups. Pacific Journal of Math., 32, No. 3 (1970), 753--766.
3. --7>-- The center and the commutator subgroup i n hopfian groups. Ark. Mat., 9, (1971), 181-- 192.
4. --*-- A conjecture on hopficity and related results. Arch. Math. (Basel), 32 (1971), 429--455.
5. --))-- Some new groups admitting essentially unique directly indecomposable decompositions.
Math. Ann., 194 (1971), 123--125.
6 .--~>-- A problem i n group theory. Amer. Math. NIonthly, 79 (1972), 379--381.
.Received September 17, 1971. 1%. Hirshon
Polytechnic Institute of Brooklyn, 333 J a y Street
Brooklyn, N.Y., IJ.S.A.
