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

90 Attila Szakács The subring K

p

X of the ring K is defined similarly. The ring K is called ^-divisible if K

v = K

. Let G{p] denote the subgroup {g £ G : g

p = 1} of G. Then the factorgroup G

x[p]/G

x+

1 [p] can be considered as a vector space over GF{p

) the field of p elements and the cardinality of a basis of this vector space is called the A-th Ulm-Kaplansky invariant f\{G) of the group G concerning to p. S. P. Novikov had raised the problem of determining the invariants of the group V*{KG) when G has a p-power order and K is a finite field of characteristic p. This was solved by A. Bovdi and the author in [1]. In [2] we gave the Ulm-Kaplansky invariants of the unitary subgroup W

P(KG) of the group V

P{KG

) whenever G is an arbitrary abelian group and K is a commutative ring of odd prime characteristic p without nilpotent elements. Here we continue this works describing the unitary subgroup W

2{KG) of the Sylow 2-subgroup V

2{KG) of the group V{KG) in case when G is an arbitrary abelian group and K is a commutative ring with unity of characteristic 2 without zero divisors. Note that for the odd primes p the problem of determining the UlmKaplansky invariants of the group W

p{KG) is based, in fact, in the following statement W

P{KG) = {x~

lx*: x £ V

P{KG)} (see [2]). But in case p = 2 this statement is not true and in the characterization of the group W

2{KG) we must keep in mind the following lemma. Lemma 1. Let G be an abelian group of exponent 2

n+

1 {n > 0) and K a commutative ring with unity of characteristic 2 without zero divisors. Then {V*(KG)y

n = G

2" . Proof. At first we shall prove the lemma for a finite group G. We shall use induction on the exponent of G. Let n = l, i.e. G is a group of exponent 4. We shall prove by induction on the order of G that {V*{KG)f = G

2

. Let G = (a

: a

4 = 1). Then the element x = ao + 0:1 a + CX.20? -f a^a

3 £ V{KG) is unitary if and only if xx* = 1 + (ao + Q

2)(«i + a

3)(a + a

3) = 1. Hence a

0 = ö2 or = 03. If ai = <23 then, according to the condition «0 + + c*2 + = 1» the unitary element x has the form x = 1 + 02(1 + a

2) + ai (a + a

3

) and x

2 = 1. If o

0 = <^2 then x = c*o(l + a

2 ) + c*i a+(l-f )°

3-