The number of distinct cycle lengths in the cycle structure of an automorphism on an abelian group

Abstract

Let G be a finitely generated free abelian group of rank n, and σ an automorphism of G. The set of distinct lengths of non-zero cycles in the cycle structure of σ is written as S(σ,G). It was proved by Klerk et al. that the structure of an arbitrary automorphism σ of G possesses at most finitely many distinct cycle lengths ([4], Proposition 3.2). A problem arises naturally: How many distinct cycle lengths possibly occur in the cycle structure of an automorphism σ of G. In this article, we prove that |S(σ,G)|≤2m if n=2m is even, or |S(σ,G)|≤2m−1 if n=2m−1 is odd, the automorphism σ for which |S(σ,G)| attains the upper bound is characterized definitely.