The Reidemeister spectrum of split metacyclic groups

TL;DR

This paper determines the Reidemeister spectrum for a class of split metacyclic groups, specifically those of the form $C_{n} times C_{p}$ with a non-trivial action, expanding understanding of automorphism classes.

Contribution

It provides a complete characterization of the Reidemeister spectrum for split metacyclic groups of the specified form, a previously unresolved problem.

Findings

Reidemeister spectrum explicitly calculated for $C_{n} times C_{p}$ groups.

Classification of automorphisms based on their Reidemeister numbers.

Enhanced understanding of conjugacy relations in split metacyclic 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 of equivalence classes for this relation is the Reidemeister number $R(\varphi)$ of $\varphi$. The set $\{R(\psi) \mid \psi \in \mathrm{Aut}(G)\}$ is called the Reidemeister spectrum of $G$. We fully determine the Reidemeister spectrum of split metacyclic groups of the form $C_{n} \rtimes C_{p}$ where $p$ is a prime and the action is non-trivial.