REPRESENTATION
METABELIAN
V.
OF
GROUPS
N.
FINITELY
BY
GENERATED
MATRICES
Remeslennikov
UDC 519.45
T h e t h e o r y of m e t a b e l i a n g r o u p s is in m a n y r e s p e c t s r e l a t e d to t h e d e v e l o p m e n t of the t h e o r y of m o d u l e s o v e r c o m m u t a t i v e r i n g s . T h u s , t h e c o n d i t i o n of m a x i m a l i t y f o r n o r m a l s u b g r o u p s of a f i n i t e l y g e n e r a t e d m e t a b e l i a n g r o u p [1] i s a d i r e c t c o n s e q u e n c e of t h e f a c t that the c o m m u t a t o r of such a g r o u p c a n be
r e g a r d e d a s a f i n i t e l y g e n e r a t e d m o d u l e o v e r a N o e t h e r i a n r i n g . T h i s r e l a t i o n s h i p i s a l s o c o n f i r m e d by the
f a c t t h a t a n y m e t a b e l i a n g r o u p c a n b e r e p r e s e n t e d b y m a t r i c e s o v e r a c o m m u t a t i v e r i n g [2]. U s i n g t h i s
f a c t , w e s h a l l p r o v e i n t h i s n o t e t h a t c e r t a i n c l a s s e s of f i n i t e l y g e n e r a t e d m e t a b e l i a n g r o u p s , for e x a m p l e
torsion-free groups, are isomorphically representable by matrices over fields.
By the l e t t e r M we s h a l l d e n o t e t h e c l a s s of f i n i t e l y g e n e r a t e d m e t a b e l i a n g r o u p s u n d e r c o n s i d e r a t i o n .
S i n c e a n y g r o u p G E M s a t i s f i e s t h e c o n d i t i o n of m a x i m a l i t y f o r n o r m a l s u b g r o u p s , i t f o l l o w s t h a t the p e r i o d i c p a r t T (G') o f t h e c o m m u t a t o r of G is a d i r e c t p r o d u c t of f i n i t e l y m a n y p r i m a r y c o m p o n e n t s of b o u n d e d
i n d e x . H e n c e f o l l o w s t h a t t h e g r o u p T (G') i s s p e c i f i e d in G ' b y a d i r e c t f a c t o r . T h e r e f o r e the g r o u p G
w i l l b e c o n t a i n e d in t h e d i r e c t p r o d u c t of f i n i t e l y m a n y g r o u p s w h o s e c o m m u t a t o r i s a p - g r o u p , p E P ( h e r e
P is t h e s e t of a l l p r i m e s and of z e r o , w h e r e a s the o - g r o u p is a t o r s i o n - f r e e g r o u p ) .
For
of g r o u p s
shall take
is a group
t h e f o r m u l a t i o n of t h e h y p o t h e s i s and o u r t h e o r e m , i t i s c o n v e n i e n t to i n t r o d u c e into M s u b c l a s s e s
Mp, P E P . A g r o u p G E Mp i f and o n l y if t h e c o m m u t a t o r of,.this g r o u p i s a p - g r o u p . If p ~ 0, we
in Mp s m a l l e r s u b c l a s s e s Mk , k = 1, 2, . . .. A g r o u p G E M~ if, and o n l y if, the c o m m u t a t o r G
of index pk
H y p o t h e s i s of M° I. K a r g a p o l o v . A g r o u p of c l a s s Mp, p E P , c a n b e i s o m o r p h i c a l l y r e p r e s e n t e d b y
m a t r i c e s of a f i e l d o f c h a r a c t e r i s t i c p , o r , in a d i f f e r e n t f o r m u l a t i o n : a f i n i t e l y g e n e r a t e d m e t a b e l i a n g r o u p
c a n be i n c l u d e d in the d i r e c t p r o d u c t o f f i n i t e l y m a n y m a t r i x g r o u p s .
O u r o b j e c t i v e i s to p r o v e t h e f o l l o w i n g t h e o r e m , w h i c h g i v e s a p a r t i a l a n s w e r to t h i s h y p o t h e s i s .
T H E O R E M 1. If a g r o u p G E M0, it c a n be r e p r e s e n t e d b y t r i a n g u l a r m a t r i c e s o v e r a f i e l d of c h a r a c teristic zero;
2) i f a g r o u p G E M~, p ¢ 0, t h e n G c a n be r e p r e s e n t e d b y t r i a n g u l a r m a t r i c e s o v e r a f i e l d of c h a r a c teristic p.
T h e p r o o f o f t h i s t h e o r e m f o l l o w s d i r e c t l y f r o m P r o p o s i t i o n s I and 2.
P R O P O S I T I O N 1. 1) i f G E M0, t h e n G c a n be r e p r e s e n t e d b y t r i a n g u l a r m a t r i c e s o v e r a N o e t h e r i a n
r i n g of c h a r a c t e r i s t i c z e r o ;
2) i f G E M~, p ~ 0, t h e n G c a n be r e p r e s e n t e d b y t r i a n g u l a r m a t r i c e s o v e r a N o e t h e r i a n r i n g of
characteristic p.
T h e p r o o f of t h i s p r o p o s i t i o n f o l l o w s d i r e c t l y f r o m : 1) t h e c o n s t r u c t i o n of a r i n g in the p r o o f of
P r o p o s i t i o n 3.1 of [2], a n d 2) the f a c t t h a t G i s a f i n i t e l y g e n e r a t e d g r o u p .
P R O P O S I T I O N 2. A N o e t h e r i a n r i n g K c a n b e i n c l u d e d in a m a t r i x r i n g o v e r a f i e l d if, and o n l y if,
t h e c h a r a c t e r i s t i c of t h e r i n g K i s e q u a l to p , p E P .
T r a n s l a t e d f r o m A l g e b r a i L o g i k a , Vol. 8, No. 1, p p . 7 2 - 7 5 , J a n u a r y - F e b r u a r y ,
a r t i c l e s u b m i t t e d J a n u a r y 17, 1969.
1969.
Original
©1970 Consultants Bureau, a division o[ Plenum Publishing Corporation, 227 West 17th Street, New York,
N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without
permission o[ the publisher. A copy o[ this article is available [ram the publisher [or $15.00.
39
Proof. It is clear that the conditions are necessary, since the characteristic
e q u a l to the c h a r a c t e r i s t i c of t h e f i e l d .
of the m a t r i x r i n g i s
L e t u s show t h a t t h e c o n d i t i o n s a r e s u f f i c i e n t . We s h a l l c o n s i d e r t h e c a s e of a r i n g o f c h a r a c t e r i s t i c
e q u a l to z e r o . In the c a s e o f a c h a r a c t e r i s t i c p ;* 0 we h a v e to m a k e o n l y m i n o r c h a n g e s in t h e r e a s o n i n g .
By the L a s k e r - N o e t h e r t h e o r e m , a r i n g K c a n b e i n c l u d e d in a d i r e c t s u m o f r i n g s K 1 x K 2 x . . .
x Kn,
w h e r e n i s f i n i t e and Ki i s a N o e t h e r i a n r i n g in which a l l t h e z e r o d i v i s o r s a r e c o n t a i n e d in a n i l p o t e n t
radical R(Ki), i = 1 .....
n. F r o m t h e p r o o f of L e m m a 2.1 o f [2] f o l l o w s t h a t t h e r i n g s K 1. . . . .
Kn c a n b e
s u c h t h a t t h e i r c h a r a c t e r i s t i c s a r e z e r o , s o t h a t w i t h o u t l o s s of g e n e r a l i t y i t c a n be a s s u m e d t h a t K c o i n c i d e s with one of the Ki, i = 1 . . . . .
n.
L e t u s c o n s i d e r a c o m p l e t e q u o t i e n t r i n g ~ f o r K. T h e n t h e r i n g ~ w i l l s a t i s f y t h e f o l l o w i n g c o n d i t i o n s :
(a) K i s a s u b r i n g of K;
(b) e a c h r e g u l a r e l e m e n t {i.e., no z e r o d i v i s o r ) of t h e r i n g K h a s a n i n v e r s e in K;
(c) e a c h e l e m e n t of t h e r i n g ~ h a s t h e f o r m ( a / b ) , w h e r e a, b E K and b i s r e g u l a r in K;
(d) K i s N o e t h e r i a n ;
(e) [R(K)] / i s a m a x i m a l and n i l p o t e n t i d e a l in K; h e r e [R(K)] / is an i d e a l , g e n e r a t e d b y R(K) in K;
(f) the c h a r a c t e r i s t i c s
of t h e r i n g s K a n d
K/[R(K)]I
a r e t h e s a m e , b e i n g e q u a l to z e r o .
T h e p r o p e r t i e s (a), (b), (c), and (d) c a n b e found in [3] (pp. 57 and 258); t h e p r o p e r t y (e) f o l l o w s
d i r e c t l y f r o m t h e f i r s t f o u r p r o p e r t i e s ; with r e g a r d to t h e p r o p e r t y (f) we c a n s e e t h a t if t h e c h a r a c t e r i s t i c
of the f i e l d K / [ R ( K ) ] I would b e p ~ 0 , t h e n p 1 E [R(K)] / , w h i c h c o n t r a d i c t s the f a c t t h a t t h e c h a r a c t e r i s t i c
o f K is e q u a l to z e r o .
L e t u s a p p l y Cohents t h e o r e m [4] to t h e r i n g K: l e t A be a c o m p l e t e l o c a l r i n g , . 7 a m a x i m a l i d e a l in
A, and l e t A h a v e the s a m e c h a r a c t e r i s t i c a s the r e s i d u e f i e l d d/..7. T h e n the r i n g A w i l l h a v e a f i e l d of
representatives L.
L e t us r e c a l l t h a t a f i e l d L is a f i e l d of r e p r e s e n t a t i v e s f o r A if f o r a c a n o n i c a l h o m o m o r p h i s m
q~ : fT~tq/:Y, w e h a v e ~f(L) ~ f4/.7, a n d t h a t t h e t o p o l o g y in A is d e t e r m i n e d by the p o w e r s o f t h e m a x i m a l
i d e a l J . It i s e a s y to v e r i f y t h a t a l l t h e c o n d i t i o n s of C o h e n ' s t h e o r e m h o l d in t h e r i n g K, and t h e r e f o r e /¢
has a field of representatives L.
S i n c e the i d e a l JR(K)]/ i s n i l p o t e n t a n d the r i n g K i s N o e t h e r i a n , it f o l l o w s t h a t K is a n a l g e b r a o f
f i n i t e r a n k o v e r L . By t a k i n g in K a b a s e , s u i t a b l e to the p o w e r s of the i d e a l [R(K)] / , we o b t a i n a r e p r e sentation of/¢ by triangular matrices over the field L.
F o r p r o v i n g the t h e o r e m i t s u f f i c e s to r e p r e s e n t t h e g r o u p G b y m a t r i c e s o v e r a N o e t h e r i a n r i n g
( P r o p o s i t i o n 1), a n d t h e n r e p l a c e the e l e m e n t s of the r i n g b y m a t r i c e s ( P r o p o s i t i o n 2).
LITERATURE
lo
2.
3.
4.
40
CITED
P . H a l l , " F i n i t e n e s s c o n d i t i o n s f o r s o l u b l e g r o u p s , " P r o c . L o n d o n M a t h . S o c . , 4, 419-436 (1954).
V. N. R e m e s l e n n i k o v , " F i n i t e a p p r o x i m a b i l i t y of m e t a b e l i a n g r o u p s , " A l g e b r a i L o g i k a , 7, No. 4,
106-113 (1968).
O. Z a r i s k i and P . S a m u e l . C o m m u t a t i v e A } g e b r a , 1, Van N o s t r a n d , P r i n c e t o n (1958). ( R u s s i a n t r a n s l a t i o n in I L , M o s c o w (1963).)
O. Z a r i s k i and P. S a m u e l , C o m m u t a t i v e A l g e b r a , 2, V a n N o s t r a n d , P r i n c e t o n (1958). ( R u s s i a n t r a n s l a t i o n in IL, M o s c o w (1963}.}