The Reidemeister spectrum of finite abelian groups

Abstract

For a finite abelian group A, the Reidemeister number of an endomorphism ϕ equals the size of Fix(ϕ), the set of fixed points of ϕ. 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 ϕ.