Certain conjugacy classes of units in integral group rings of metacyclic groups

Abstract

Let G be the metacyclic group of order pq given by G = <σ, τ: σ^p = 1 = τ^q, τστ^- = σ^j> where p is an odd prime, q ≥ 2 a divisor of p - 1, and where j belongs to the exponent q mod p. Let V denote the group of units of augmentation 1 in the integral group ring Z G of G. In this paper it is proved that the number of conjugacy classes of elements of order p in V is (p - 1)^q-1 μ₀ H/vq where ν, μ₀, and H are suitably defined numbers.