The Reidemeister spectrum of finite abelian groups

Abstract

For a finite abelian group $A$, the Reidemeister number of an endomorphism $\phi$ equals the size of $\mathrm{Fix}(\phi )$, the set of fixed points of $\phi$. 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 $\phi$.

20 pages