# Automorphisms of metacyclic groups

Research paper by Haimiao Chen, Yueshan Xiong, Zhongjian Zhu

Indexed on: 07 Sep '18Published on: 07 Dec '17Published in: Czechoslovak Mathematical Journal

#### Abstract

A metacyclic group H can be presented as 〈α,β: αn = 1, βm = αt, βαβ−1 = αr〉 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.