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

Let G be the metacyclic group of order pq given by

$\text{G = 〈σ, τ: σ}{}^{\mathrm{p}}\text{= 1 = τ}{}^{\mathrm{q}}\text{, τστ}{}^{\mathrm{?}}\text{= σ}{}^{\mathrm{j}}\text{〉}$

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 $\text{Z}$G of G. In this paper it is proved that the number of conjugacy classes of elements of order p in V is

$\text{(p ? 1)}{}^{\mathrm{q?1}}\text{μ}{}_0\text{H}\text{vq}$

where ν, μ₀, and H are suitably defined numbers.