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 ... 89 follows. If we multiply (1) by ( g - l) v~ l = 1 + g + • • • g p~ l, then we get the équation 1 + g + • • • + g v~ l — (1 + g + • • • + g p~ l )z. Hence (2) (l + g+--- + g?í)(z-l) = 0. Suppose that g ^ G n+i. Since the support of the element z—1 belongs to the subgroup G n+\ and the elements of the group G n+i belong to the cosets of the group G n by the subgroup (g), it follows that z — 1. According to the statement 1 = 1 + (g - iy = (1 + g - 1)(1 - (g - 1) + (g - l) 2 - • • • + (g - l)^ 1 ) it is easy to prove that (3) g' 1 - 1 = -(a - i) + (g - i) 2 - • • • + (g - í)'" 1. Ifin the équation (1) the element g~ l — 1 is substituted by the right side of (3) and the obtained équation is multiplied by (g — l) p~ 2, then we get (u - a)fl( 1 + g + • •. + g p~ l) = (a- u)a( 1 + g + • • • + g p~ l). Hence (u — a) = —{y — a) and this is impossible in a ring of characteristic p > 2 whenever a / v. Now let g G G n+i. The element y = z — 1 can be presented in the form y - z - 1 - ziui + h z su s where Z{ G K n(g) and U{ (i = 1,...,5) are the représentatives of the cosets of the group G n +1 by the subgroup (g). Then, according to (2), every Z{ (i — 1,s) belongs to the fundamental ideal of the ring K n(g) and hence it can be written in the form Z{ = Oi\{g — 1) + • • • + o: p_i(g — l) p_ 1. Therefore (4) Z = 1 + y i(g - 1) + • • • + y p-i(g - 1 y~ l where the support of the elements yi (i = 1,... — 1) consists of the représentatives of the cosets of the group G n+1 by the subgroup (g). Ein the équation (1) the elements g' 1 — 1 and 2 are substituted by the expressions shown in (3) and (4), and the obtained équation is multiplied by ( g — l) p~ 2, then we get 2 (i/ - a)a( 1 + g + • • • + g v~ X) = Vi{l + g + • • • + ^1).
