Abstract UNITARY SUBGROUP OF INTEGRAL GROUP RINGS

a-1, da E A. We find generators up to finite index of the unitary subgroup of 7LG. In fact, the generators are the bicyclic units. For an arbitrary group G, let B2(7LG) denote the group generated by the bicyclic units. We classify groups G such that B2(7LG) is unitary. Let 7LG be the integral group ring of an arbitrary group G and let f: G-> U(7L) = {±1} be an orientation homomorphism. For each x = 1: agg, we put xf = ag f(g)g-1. In particular, if f is trivial, gEG xf coincides with the standard x *. Let U(ZG) be the group of units of 7LG. Then u E U(ZG) is called f-unitary if u-l = uf or u-1 =-uf. All f-unitary elements of U(ZG) form a subgroup Uf (7LG) containing G x U(7L). We refer to Uf (ZZG) as the f-unitary subgroup of U(ZZG). Interest in the group Uf (7LG) arose in algebraic topology and unitary K-theory [4]. We are interested in the constructive description of Uf (ZLG). If G is finite cyclic, then Bovdi [1] gave a linearly independent set of generators for a torsion free subgroup of finite index in Uf (7ZG). This was extended to finite abelian groups by Hoechsmann-Sehgal in [3]. We give generators up to finite index of Uf (7ZG) if G is a finite dihedral group. In fact, the generators consist of the bicyclic units. The subgroup B2(7ZG) of U(ZG) generated by all the bicyclic units of 7LG plays an important role in the study of U(ZG) (see [5], [6]). In Theorem 2, we characterize groups G for which B2 (7LG) is unitary