Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1994. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 22)

SZAKÁCS, A., Unitary subgroup of the Sylow p-subgroup of the group of normalized units in an infinite commutative group ring

92 Attila Szakács belong to distinct cosets of the group V

p

n[p] by the subgroup V

pn+

1\p], Indeed, suppose that x

a = x

cz for distinct elements a and c from the set 7r

2 and for some z G V

p [p]. Then (1 + c(l + v)(g - 1)) (1 + a1(l + tT

1)^

1 - 1)) = = (1 + a(l + v)(g - t))'

1 (1 + c~\ 1 + v~

l)(g'

1 - 1)) s. If we multiply the équation (6) by (g — l)

p_

1

, then it follows that (1 -f g + ' • • + g

v~

l){

z — 1) = 0 and we can write z in the form (4). Let us now multiply the équation (6) by (

g — l)

p~

2

. Then, by useing (3) and (4), we get (c - a)( 2 + t; + tr

1)^ + g + • • • + g

p~

l

) = îft (1 +g + --- + g

p~

l). Since a and c are from distinct cosets of the group G

n by the subgroup (g,v), it follows that c and cv belong to the support of the left side of this équation. Hence they coincide with some elements from the support of the right side, which belongs to the subgroup G

n+\. Therefore c G G

n+i and cv G G

n+1, but this contradicts the choice of v £ G

n+iTherefore, the case C) is fully considered and the theorem is proved for a finite ordinal u> = n. Let us consider the case of infinite ordinal u. Let w be an arbitrary infinite ordinal R = K^^H = G

U

} ^ G

u+i and the Sylow p-subgroup of the group G

u is not singular. Then W(KG)

p

U Ç W(RH) C V

P(RH) and by transfinite induction it is easy to prove (7) (V

p(KG)f = Vp(RH). For the group V

P(RH

) we can construct the set M as in the above shown cases A), B) and C). Since in each of these cases the set M consists of elements of form x = y~

ly* and, by (7), y belongs to the group V

v(RH) = íjj (Vp(KG)) , it follows that the elements x are the représentatives of the cosets of the group W

p (KG)[p] by the subgroup W

p (KG)\p

]. Therefore, for an arbitrary infinite ordinal u the Ulm-Kaplansky invariants of the group W{KG) can be calculated in the above shown way for the case a

: — n.