Torsion units of integral group rings of metacyclic groups

Abstract

Let VZ G (respectively, VQ G) denote the group of units of augmentation 1 in the integral (respectively, rational) group ring of a finite group G. It has been conjectured [[7.]] that each element of finite order of VZ G is conjugate in VQ G to an element of G (see also [1.] and [6.]). To the best of our knowledge, the only nonabelian case (other than the Hamiltonian 2-groups) where this conjecture has been verified is G = S₃ [[5.], 529-534]. In this paper this conjecture is verified for the metacyclic group G = < σ , τ: σ^p = 1 = τ^q, τστ^-1 = σ^j> (p, q primes, p = 1 mod q, j^q = 1, j N= 1 mod p) by expressing VZ G and VQ G as semidirect products of groups of q × q matrices. Although S. Galovitch, I. Reiner, and S. Ullom [Mathematika. 19 (1972), 105-111] obtained a description of VZ G, the discussion of torsion units was not attempted by them.