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Automorphisms of metacyclic groups

Haimiao Chen, Yueshan Xiong, Zhongjian Zhu

Abstract

A metacyclic group $H$ can be presented as $\langle α,β\mid α^{n}=1, \ β^{m}=α^{t}, \ βαβ^{-1}=α^{r}\rangle$ for some $n,m,t,r$. Each endomorphism $σ$ of $H$ is determined by $σ(α)=α^{x_{1}}β^{y_{1}}, σ(β)=α^{x_{2}}β^{y_{2}}$ for some integers $x_{1},x_{2},y_{1},y_{2}$. We give sufficient and necessary conditions on $x_{1},x_{2},y_{1},y_{2}$ for $σ$ to be an automorphism.

Automorphisms of metacyclic groups

Abstract

A metacyclic group can be presented as for some . Each endomorphism of is determined by for some integers . We give sufficient and necessary conditions on for to be an automorphism.

Paper Structure

This paper contains 5 sections, 7 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Determining all automorphisms
  3. Preparation
  4. The method
  5. Main result

Key Result

Lemma 2.1

If $s>1$ with $\deg_{p}(s-1)=\ell\ge 1$ and $x>0$ with $\deg_{p}(x)=u\ge 0$, then

Theorems & Definitions (15)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Remark 2.6
  • ...and 5 more