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

Unitary subgroup of the Sylow 2-subgroup of the group of. 95 of the set II. Since G

n is infinite, it is easy to see that \G

n\ = |XX| = max {infills |}. Let us suppose at first that \G

n\ = j Hi j

. Without loss of generality we can assume that the representative of the coset a1 (g) is the element a1. Let E denote the set which has a unique representative in every subset of the form {a, a"

1} C Iii and y

a = 1 + a(l +g + • • • + g

q~

l). Then \G

n\ = \E\ and the elements of the set M = {x

a = y

ay

a* = 1 + (a + cr

l)(l + g + • • • + g

q~

l)

: a £ E} belong to the different cosets of the group V

2" [2] by the subgroup V

2 [2]. Indeed, from the supposition x

a £ V

2"

+

1[2] it follows that ag

l £ C

n

+i for every i = 0,1,..., q — 1, but this contradicts to the choice of the element g £ G

n \ G

n+1. It is easy to see that x

a is a unitary element and so x

a £ W

2 [2]\W

2

n* [2]. Supp ose that a and c are the distinct elements of the set E. If x

a = x

cz for some z £ V

2 then 2 = x

ax

c* = l + (a-ha~

1 +c + cl)(l+g + --- + g

q1). According to the choice of the elements of the set E we have that the elements a,a1,c, c~

l belong to the distinct cosets. of the group G

n by the subgroup (g). Hence from the condition 2 £ V

2 it follows that a £ G

n+1, a9 ^ G

n+i, which contradicts to the choice of the element g £ S

n\ S

n+\. Let be now \G

n\ = |n

2|. If G

2 = 1 then W(KG) = V(KG) and f

0(W) = fo{V

2) = \G\. If n > 0 and G

n+

i = 1 then, by Lemma 1, fn(W) = f

n{G). Suppose that G

n+i / 1. Then the group G

n has such element v of order not equals to 2 that (g) fl (v) = 1. If a such representative of the coset a(g) that a

2 £ (g) and a

2 ^ 1, then a

2 = g

% £ G

n

+1 and, according to the choice of the element g, the integer i is divisible by 2. In this case in role of the representative of the coset a(g) in the set II2 we can choose the element a\ = ag~%. Therefore, we can assume that the set n

2 consists of the elements of second order. Since (g) fl (v) = 1, it follows that from the II

2 we can choose a subset II

2 which elements belong to the distinct cosets of the group G

n by the subgroup (g

:v) and \G

n\ = |n

2|. Let y

a = 1 + av( 1 + g + • • • + g

q~

l). Then the set M = {x

a = y

ay

a* = + + g

q~

l)

: a £ fi

2

} has the need property. Indeed, the cosets x

a V

2 and x

cV

2 coincide if and only if = 1 + (a + c)(v + v~

l )(1 + g + • • • + g

q~

l) £ F

2"

+

1