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 of the group of normalized units in an infinite commutative group ring ATTILA SZAKÁCS Abstract. Let G be an abelian group, K a ring of prime characteristic p and let V(KG) denote the group of normalized units of the group ring KG. An element U — ag9 ^ V(KG) is called unitary if I/1 coincides with the element U* = X/06G • The set of all unitary elements of the group V(Ií G) forms a subgroup VJKG). S. P. Novikov had raised the problem of determining the invariants of the group K(Ä G) 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. Here we give the invariants of the unitary subgroup of the Sylow J9-subgroup of V(KG) whenever G is an arbitrary abelian group and Ií is a commutative ring of odd prime characteristic p without nilpotent elements. 1. Introduction Let G be an abelian group, K a ring of prime characteristic p and let V(KG) denote the group of normalized units (i.e. of augmentation 1) of the group ring KG. We say that for x = ^ a

gg G KG the element geG x* = ^^ ag9~

l is conjugate to x, and if x* = x, then x is selfconjugate. geo It can be seen that the map x —> x* is an anti-isomorphism (involution) of the ring KG. An element u G V{KG) is called unitary if u~

l = u*. The set of all unitary elements of the group V(KG) obviously forms a subgroup, which we therefore call the unitary subgroup of V{KG), and we denote it by V*(KG). Let G

v denote the subgroup {g

p \ g G G] of p-th powers elements of G and u an arbitrary ordinal. The subgroup G

v of the group G is defined by 0 transfinite induction in following way: G

p = G, for non-limited ordinals Research supported (partially) by the Hungárián National Foundation for Scientific Research (OTKA), Grant No. T014279.