The Reidemeister spectrum of finite abelian groups

TL;DR

This paper characterizes the Reidemeister spectrum of finite abelian groups by identifying which divisors of the group order occur as Reidemeister numbers of automorphisms, providing bounds on fixed points.

Contribution

It fully determines the Reidemeister spectrum of finite abelian groups and establishes bounds on fixed points of automorphisms related to a given automorphism.

Findings

Reidemeister spectrum corresponds to divisors of group order

Complete characterization of automorphism fixed points spectrum

Provides bounds on fixed points of automorphisms

Abstract

For a finite abelian group $A$, the Reidemeister number of an endomorphism $\varphi$ equals the size of $\mathrm{Fix}(\varphi)$, the set of fixed points of $\varphi$. Consequently, the Reidemeister spectrum of $A$ is a subset of the set of divisors of $|A|$. We fully determine the Reidemeister spectrum of $|A|$, that is, which divisors of $|A|$ occur as the Reidemeister number of an automorphism. To do so, we discuss and prove a more general result providing upper and lower bounds on the number of fixed points of automorphisms related to a given automorphism $\varphi$.