Functions realising as abelian group automorphisms

TL;DR

This paper characterizes functions on sets and abelian groups that can serve as automorphisms under a compatible group operation, extending to infinite groups isomorphic to integer lattices.

Contribution

It provides necessary and sufficient conditions for functions to be automorphisms of groups formed from arbitrary sets and specific abelian groups, including infinite cases.

Findings

Conditions for functions to be automorphisms of cyclic groups.

Extension of results to abelian groups of order p^2.

Complete characterization for countably infinite groups isomorphic to Z^n.

Abstract

Let $A$ be a set and $f:A\rightarrow A$ a bijective function. Necessary and sufficient conditions on $f$ are determined which makes it possible to endow $A$ with a binary operation $*$ such that $(A,*)$ 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 $(A,*)$ isomorphic to ${\Bbb Z}^n$ for some $n\ge 1$, and with $f\in \mbox{Aut} (A)$, are completely characterised.