When is a permutation of the set $\Z^n$ (resp. $\Z_p^n$, $p$ prime) an automorphism of the group $\Z^n$ (resp. $\Z_p^n$)?

Authors: BEN-EBEN DE KLERK, JOHAN H. MEYER

Abstract: For a given positive integer $n$, the structure, i.e. the number of cycles of various lengths, as well as possible chains, of the automorphisms of the groups $(\Z^n, +)$ and $(\Z_p^n,+)$, \ $p$ prime, is studied. In other words, necessary and sufficient conditions on a bijection $f : A \ra A$, where $|A|$ is countably infinite (alternatively, of order $p^n$), are determined so that $A$ can be endowed with a binary operation $*$ such that $(A,*)$ is a group isomorphic to $(\Z^n,+)$ (alternatively, $(\Z_p^n,+)$) and such that $f\in \Aut(A)$.

Keywords: Automorphism, abelian group

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