Automorphisms of metacyclic groups

Abstract

A metacyclic group H can be presented as 〈α,β: αⁿ = 1, β^m = α^t, βαβ⁻¹ = α^r〉 for some n, m, t, r. Each endomorphism σ of H is determined by $\sigma(\alpha)=\alpha^{x_1}\beta^{y_1}, \sigma(\beta)=\alpha^{x_2}\beta^{y_2}$ for some integers x₁, x₂, y₁, y₂. We give sufficient and necessary conditions on x₁, x₂, y₁, y₂ for σ to be an automorphism.