The Reidemeister spectrum of ZM-groups

TL;DR

This paper determines the Reidemeister spectrum of ZM-groups by analyzing automorphisms and their conjugacy classes, extending previous results in the field.

Contribution

It provides a complete characterization of the Reidemeister spectrum for ZM-groups, a class of finite groups, which was not previously known.

Findings

Reidemeister spectrum of ZM-groups is explicitly determined.

Extension of previous results on automorphism conjugacy classes.

Provides a foundation for further studies on Reidemeister numbers in finite groups.

Abstract

Given a group $G$ and an automorphism $\varphi$ of $G$, two elements $x,y\in G$ are said to be $\varphi$-conjugate if $x=gy\varphi(g)^{-1}$ for some $g\in G$. The number $R(\varphi)$ of equivalence classes with respect to this relation is called the Reidemeister number of $\varphi$ and the set $\{R(\varphi)|\varphi\in {\rm Aut}(G)\}$ is called the Reidemeister spectrum of $G$. In this paper, we determine the Reidemeister spectrum of ZM-groups, extending some results of \cite{13}.