Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1997. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 24)
SZAKÁCS, A., Unitary subgroup of the Sylow 2-subgroup of the group of normalized units in an infinite commutative group ring
Unitary subgroup of the Sylow 2-subgroup of the group of normalized units in an infinite commutatuve group ring ATTILA SZAKÁCS Abstract. Let G be an abelian group, K a commutative ring with unity of prime characteristic p and let V(KG) denote the group of normalized units of the group ring KG. An element u—^2 e G a gg£V(KG) is called unitary if u1 coincides with the element = ^ e G ot gg~ 1. The set of all unitary elements of the group V(KG) forms a subgroup V+(KG). 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 problem was solved by A. Bovdi and the author. We gave the Ulm-Kaplansky invariants of the unitary subgroup of the Sylow p-subgroup of V(KG) whenever G is an arbitrary abelian group and K is a commutative ring with unity of odd prime characteristic p without nilpotent elements. Here we continue this works describing the unitary subgroup of the Sylow 2-subgroup of the group V(KG) in case when G is an arbitrary abelian group and K is a commutative ring with unity of characteristic 2 without zero divisors. Let G be an abelian group and K a commutative ring with unity of prime characteristic p. Let, further on, V(KG) denote the group of normalized units (i.e. of augmentation 1) of the group ring KG and V p(KG ) the Sylow p-subgroup of the group V{KG). We say that for x = ag9 ^ ^ G the element z* = ag9~ l conjugate to x. Clearly, the map x —* x* is an anti-isomorphism (involution) of the ring KG. An element u E 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 p denote the subgroup {g p : g E G } and A an arbitrary ordinal. The subgroup G p of the group G is defined by transfinite induction in following way: G p° = G, for a non-limited ordinals A is a limited ordinal, then G p X = fl t/ <A G p" . * Research (partially) supported by the Hungarian National Research Science Foundation, Operating Grant Number OTKA T 16432 and 014279.
