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

Unitary subgroup of the Sylow p-subgroup ... 87 the map, defined in following way: ip(x) = x~

lx*(x G V

p(KH)). Then íj) is an endomorphism of the group V

p(K H), ip(V

p(KH)) Ç W(KH) and the kernel of tp coincides with the subgroup C(KH). Hence the index of the group V

p(KH) by the subgroup C(KH) not greater than the order of the group W(KH). Since C(KH)[\W(KH) = 1, it follows that this index coincides with the order of the group W(KH), and so V

p(KH) = C(IíH) X W(KH). The statements are proved. It is easy to prove the following statements (see [2]): 1) \K

P\ = \K\\ 2) if n is a nonnegative integer and J(G

P [p]) is the ideal of the ring (KG)

P , generated by the elements of the form g — 1 (g G G

p

n\p\), then V

p

n(KG)\p\ = V(K

nG

n)[p] = 1 + J(G

p

n\p}). First we shall prove the theorem for a finite ordinal u = n. Suppose that n is a nonnegative integer, the Sylow p-subgroup S

n of the group G

n is not singular, G

n / 1 and at least one of the ordinals \K

n\ or \G

n\ is infinite. Since W

pn+

1\p] Ç W

p

n\p] Ç = V(K

nG

n), it follows that fn(W) < < max{|/í

n|, |G„|} = ß. In the proof of the équation f

n(W) = ß we shall consider the following cases: A) \K

n\ > \G

n\, B) I

G

nI > \K

n\ and S

n / S

n+

U C) \G I > n| and S

n — <S'n+1j and in each of this cases we shall construct a set MC W

p (KG)[p] of cardinahty ß = max{|Ä

r

n|, |G

n|} elements of which belong to différent cosets of the group V

p

U {KG)\p] by the subgroup V

pU

+' (KG)\p\. This will be sufficient for the proof of the theorem, because the elements of a set M constructed in this way can be considered as the reçresentatives of the cosets of the group W

p (KG)[p] by the subgroup W

p (I(G)[p]. Note that we will choose the elements of the set M in form y~

1y* (y G V

p (KG)). Suppose A) holds, i.e. \K

n\ > \G

n\. It is easy to prove that in this case the Sylow p-subgroup S

n of the group G

n has an element g of order p and there exists an a G G

n such that one of the following conditions holds: ^i) {g

), a £ (g) and a

2 $ {

g), M) G

n / (g), a £ {g) and a

2 G (g),