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

88 Attila Szakács A3) G

n = (g), and in cases Ai) and A

2) one of the elements a or g does not belong to the subgroup G

n

+i

. Indeed, if g G G

n+

1 then, by the condition G

n ^ G

n+

1, the set G

n \ G

n+

1 has the required element a. Suppose Ai) holds. Let 0 be a nonzero element of the ring K

n and y

a = 1 — aa( 1 -f g + • • • -f g

p~~

1)- We shall prove that the set M = {x

a = Va'

1 Va* = 1 + a(a - a"

l)(l + g + . •. + g

v~

l) \ 0/û G Kn} has the above declaxed property. Indeed, since a

2 ^ {g

), it follows that the elements a and a~

l belong to différent cosets of the group G

n by the subgroup (g). Hence x

a / 1. It is easy to see that x

a* = 1- a(a - a1)(l + g + • • • + g

p~

l) = and x

n

p — 1. Therefore X (y IS cl unitary element of order p of the group V(K

nG

n). If x

a G V

p then, from the condition a

2 £ (g

), it follows that ag

l G G

n+i for every i = 0,1,... ,p— 1. Hence the elements a and ag belong to the group G

n+

1, but this contradicts the choice of elements a and g. Therefore x

a G W

p" [p] \ W

pU+

1 [p]. Suppose that the cosets x

a V

p \p

] and x„V

p [p] coincide for some différent a and u from K

n. Then x

a = x

uz for a suitable 2 G V

pH+

1

. Since x

u* = it follows that z = x

ax

v* = 1 + (a - v)(a - a

_

1)(l + g H h g

p~

l) = x

av and x

a_

v belongs to the subgroup V

pn+

1

, which contradicts the proved above. Obviously \M\ = \K

n|

. Therefore, the constructed set M has the above declared property. Suppose now that A

2) holds. Then y

a — 1 + c^o,(g — 1) is not a selfconjugate element in the group V

p [p] \ V

p \p] and the set M can be choosen in the following way: M = {x

a = ya~

ly** I 0 / a G K

n} . It is easy to prove that X Q- — CC q/ ^ S O j from the assumption x

a = x„z (o: / z G V

p ) the équation (l + va(g-l)){l + aa~

l{g1 - 1)) = = (1 + oca{g - 1)) (1 + ua-'ig1 - 1)) 2: