Torsion units of integral group rings of metacyclic groups

Abstract

Let V Z G (respectively, V Q 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 [ H. Zassenhaus, in “Studies in Mathematics,” pp. 119–126, Instituto de Alta Cultura, Lisbon, 1974 ] that each element of finite order of V Z G is conjugate in V Q G to an element of G (see also R. K. Dennis [“The Structure of the Unit Group of Group Rings,” Lecture Notes in Pure and Applied Mathematics Vol. 26, Sect. 8, Dekker, New York, 1977] and S. K. Sehgal [“Topics in Group Rings,” Problem 23, Dekker, New York, 1978]). To the best of our knowledge, the only nonabelian case (other than the Hamiltonian 2-groups) where this conjecture has been verified is G = S 3 [ I. Hughes and K. R. Pearson, Canad. Math. Bull. 15 (1972) , 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 V Z G and V Q 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 V Z G, the discussion of torsion units was not attempted by them.