Functions realising as abelian group automorphisms

Abstract

Let $A$ be a set and $f:A\to A$ a bijective function. Necessary and sufficient conditions on $f$ are determined which makes it possible to endow $A$ with a binary operation $\ast$ such that $(A,\ast )$ is a cyclic group and $f\in \mbox{Aut}(A)$. This result is extended to all abelian groups in case $|A|=p^2,p$ a prime. Finally, in case $A$ is countably infinite, those $f$ for which it is possible to turn $A$ into a group $(A,\ast )$ isomorphic to ${\mathbb{Z}}^n$ for some $n\ge 1$, and with $f\in \mbox{Aut} (A)$, are completely characterised.

17 pages