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 ... 91 Indeed, from the supposition x

a = x

cz (

z G V

pU+

1 the équation (l + c(

f

f-l))(l + a1(

f

f1-l)) = = (1 + i(g - 1)) (l + c~

1(g~

1 - 1)) z follows. Multiplying the équation (5) by (

g - 1)

9_

1 we get the statement (1 + 5 + 1- 9

q~

l) = (1 + g + h g

q~

l )z- As in above, we can prove that from this équation and the condition g ^ G

n+i the statement z = 1 follows. Substituting the element g — 1 in (5) by the right side of (3) and the obtained équation is multiplied by (g —1)

9~

2 we have that 2(c — a)(l-\-g-\ Vg

q~

l) — 0. But it contradicts the fact that a and c belong to distinct cosets of the group G

n by the subgroup (g). So the case B) is fully considered. Suppose C) holds , i. e. \G

n\ > \K

n\ and the Sylow p-subgroup S

n of the group G

n is p-divisible. Let us fix an element g G S

n\p] and choose v G G

n \ G

n+i such that p does not divide the order of v. Since \S

n\ = [S

n : (g)] > |{r)| and v S

n, it follows that the caxdinality of the set 7

r = ír (G n / (g, v)) coincides with \G

n\. Obviously the set ír décomposés to two disjoint subsets 71^ = {a G 7r | a

2 £ {v,g)} and tt

2 = {a G TT I a

2 G (v,g)}. Let \G

n\ = IxJ, \l + v, ifv

2 = l, E be a set which has a unique représentative in every subset of the form {a, a~

l \ a E 7T

1} and y

a = 1 — av( 1 + g + • • • + g

p~

l ). Then the set M can be choosen in the following way: M = {x

a = y

aly

a* = 1 + {a-a~

1)v{l + g + --- + g

p1) | a G E}. Indeed, from the équation x

a = x

cz (z G V

p + follows that 0 = 1 + (a - a"

1 - c + c1)ïï(l + g + • • • + g*'

1) G . Hence, according to the construction of the set E

, the elements a and av belong to the group G

n+\

, but this contradicts the condition v (fc G

n+1

. Assume \G

n\ = |7T

2|. The elements of M = {x

a = (1 + a( 1 + v)(g - l))"

1 (l + a~

l(l + TT

1)^

1)) I« £ TT

2}