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

90 Attila Szakács Hence the element a from the support of the left side of this équation coincides with some element from the support of the right side. But this contradicts the condition a £ G

n+i, since the support of the right side belongs to the group G

n+1. Therefore, the case A

2) is completed. Suppose A3) holds, i.e. G

n = {

g). Then G

n+\ = 1. As in the previous case it is easy to prove that the set M = [x

a = (1 + a(g - I))"

1 (1 + a(g~

l - 1)) | 0 / a 6 K

n) has the needed property. Therefore, the proof is complété whenever A) holds. Suppose now that B) holds, i.e. \G

n\ > \K

n\ and the Sylow p-subgroup S

n of the group G

n does not coincide with the Sylow p-subgroup S

n+1 of the group G

n+1. Then the set S

n \ S

n+1 has an element g of order q = p

r. Let 7

r = 7T(G

n/(g)) denote the füll set of représentatives of the cosets of the group G

n by the subgroup (g). Consider two disjunct subsets ir

1 = {a G 7T I a

2 £ (g)} and 7t

2 = {a G 7r | a

2 G (^)} of the set 7r.lt is easy to see that \G

n\ = |TT| = maxl

^J, |7T

2|}. Let us suppose first that \G

n\ = 17r

31. Without loss of generahty we can assume that the représentative of the set a1 (g) is the element a1. Let E denote a set which has a unique représentative in every subset of the form {a, a~

l I a G TTJ and y

a = 1 - a~

l (1 + g + • • • + g

q~

l ). Then \G

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

a = y

aly

a* = 1 + (a"

1 - a){ 1 +g + --- + g

q~

l) \ a G E} belong to différent cosets of the group V

p

n[p] by the subgroup V

pn+

1[p\. Indeed, it is easy to see that x

a G W

p

n[p] \ W

?

n \p]. Suppose that a and c are distinct elements of the set E. If x

a = x

cz for some 2 G V

p

n , then 2 = x

ax

c* = 1 + (

a~

l - a - c1 + c)(l + g + • • • + g

q~

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

, c, c

_

1 belong to distinct cosets of the group G

n by the subgroup (g). Hence from the condition z G V

p it follows that ag

l G G

n+

1 (î = 0,1, • • •, ç — 1), which contradicts the choice of the element g G S

n\ S

n+\. Let be |

G

n\ = |7r

2|- Then the set M can be choosen in the following way: M = {x

a = (1 + a(g - I))1 (1 + a~

l(g~

1 - 1)) | a G TT

2}.