Inverse ambiguous maps on infinite groups

Abstract

Let $G$ be a group. A bijection $f\colon G\to G$ is called inverse ambiguous if $f^{-1}(x)=f(x)^{-1}$ for every $x\in G$. We prove that every infinite group admits an inverse ambiguous function. An inverse ambiguous automorphism, however, can occur only on an abelian group. For a finitely generated abelian group $A\cong\Z^r\oplus T$, with $T$ finite, we show that $A$ admits an inverse ambiguous automorphism if and only if $r$ is even and $T$ admits one. This leads to an explicit classification by combining the result with Toborg's finite classification. We also prove that an infinite locally finite group is abelian if each of its proper subgroups admits an inverse ambiguous automorphism, and we give examples showing that both hypotheses are necessary. Finally, we prove that a residually finite group is abelian whenever all its finite quotients admit inverse ambiguous automorphisms. We also construct an infinite residually finite abelian group whose finite quotients all admit inverse ambiguous automorphisms, although the group itself admits none.