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
86 Attila Szakács (that is if u = v + 1): G v = yG p J , and if CJ is a limited ordinal, then G p" = P| G p U . The subring K p of the ring K is defined similarly. The ring Ii is called ^-divisible if K p = Ii. Let G\p] denote the subgroup {g E G \ g p = 1} of G. The factorgroup G p" \p]/G pUJ+ 1 \p\ can be considered as a vector space over GF{p) the field of p elements, and the cardinality of a basis of this vector Space is called the cj-th Ulm-Kaplansky invariant f u(G) of the group G concerning p. S. P. Novikov had raised the problem of determining the invariants of the group V*(KG) when G has a p-power order and K is a finite field of characteristic p. This was solved by A. Bovdi and the author in [1]. Here we continue this work by giving the Ulm-K aplansky invariants of the unitary subgroup W(KG ) of the Sylow p-subgroup V P(KG ) of V(KG) whenever G is an arbitrary abelian group and Ii is a commutative ring of odd prime characteristic p without nilpotent elements. Theorem. Let eu be an arbitrary ordinal, Ii a commutative ring of odd prime characteristic p without nilpotent elements , P the maximal divisible subgroup of the Sylow p-subgroup S of an abelian group G, G u = G p , S u = and K u = K p W . Let, further on, V v = V p(KG) denote the Sylow p-subgroup of the group V = V(KG ) of normalized units in the group ring KG and W = W{KG) the unitary subgroup ofV p(KG). In case P / 1 we assume that the ring K is p-divisible. If G u ^ S u ^ 1 and at least one of the ordinals \KJ\ or is infinite, then the u-th Ulm-K aplansky invariant / w(W) of the group W concerning p equals fu(W) = UV P) = max{|G w|, \K„\}. PROOF. Note that if G u = or S u = 1 then, according to [2], f u(V p) = 0 and hence f u(W) = 0. Let C(KG) denote the subgroup of selfconjugate elements of the group V P(KG). Then the following statements are true: V P(KG) = C(KG ) x W(KG) and W(KG) = {x~ lx * I x E V P(KG)}. Really, if x £ C(KG) f| W(KG), then x = x* and xx* = 1. Hence x 2 = 1 and since p > 2, it follows that x = 1. Therefore, C(KG) X W(KG) is a subgroup of V P(KG). Let H be a finite subgroup of the group G and tp
