Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1997. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 24)

SZAKÁCS, A., Unitary subgroup of the Sylow 2-subgroup of the group of normalized units in an infinite commutative group ring

92 Attila Szakács Theorem. Let X be an arbitrary ordinal

, K a commutative ring with unity of characteristic 2 without zero divisors, P the maximal divisible subgroup of the Sylow 2-subgroup S of an abelian group G, G

x = G

2 , S\ = S

2 , K\ = K

2 . Let, further on, V

2 = V

2(KG) denote the Sylow 2-subgroup of the group V = V(KG

) of normahzed units in the group ring KG and W = W(KG) the unitary subgroup of V

2(KG). In case ? / 1 we assume that the ring K is 2-divisible. If G\ / G\+1, S\ ^ 1 and at least one of the ordinals |K\| and |Ga| is infinite, then the X-th TJlm-Kaplansky invariant f

x(W) of the group W concerning to 2 is characterized in the following way: ( max{|G|, I A' I

}, if X = 0, fx(W) = i f

x(V

2) = max{|G

A|, \K

X\}, if X > 0 and G

x+

x / 1, ifx{G), ifX>0andG

x+

l =1. Proof. It is easy to prove the following statements (see [3]): 1) \K

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

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

2 generated by the elements of the form g—1 (g £ C

r [2]), then V

2* (KG)[2] = V(K

nG

n)[2] = l + J(G

2

n[2}). Note if G

x = GA+i or 5a = 1 then, according to [3], f

X(V

2) = 0 and hence f

x(W) = 0. At first we shall prove the theorem for a finite ordinal A = n. Suppose that n is a nonnegative integer, the Sylow 2-subgroup S

n of the group G

n is not singular, G

n ^ G

n+i and at least one of the ordinals |Ä'

n| and \G

n\ is infinite. Since ^

2"[2]CV

2" = V(K

nG

n), it follows that fn(W) < \V

2

n I < max{|Ä'

n|, |G

n|} = ß. In the proof of the equation f

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

n\ > \G

n\, B) IG

nI > IK

nI and S

n / 5

n+

i, C) \G

n\ > IK

nJ and S

n = 5

n

+i, and in each of this cases we shall construct a set M C W

2 (KG)[2] of cardinality ß = max{|/v

n|, \G

n\] (if, keeping in mind Lemma 1, it is possible) which elements belong to the different cosets of the group V

2 (KG)[

2] by the subgroup V

2n+

1 (KG)[2]. This will be sufficient for the proof of the lemma,