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 can exist only on an abelian group. We classify the finitely generated and divisible abelian groups admitting such automorphisms. If $A\cong\Z^r\oplus T$, where $T$ is finite, then $A$ admits an inverse ambiguous automorphism if and only if $r$ is even and $T$ admits one. Together with Toborg's finite classification, this gives an explicit classification of the finitely generated abelian groups admitting such automorphisms. For a divisible abelian group $A\cong\mathbb Q^{(\kappa_0)}\oplus \bigoplus_p C_{p^\infty}^{(\kappa_p)}$, such an automorphism exists if and only if none of $\kappa_0$, $\kappa_2$, or $\kappa_p$ for $p\equiv3\pmod4$ is a finite odd cardinal. We also prove that if every proper subgroup of an infinite locally finite group $G$ admits an inverse ambiguous automorphism, then $G$ admits one that leaves every subgroup invariant. Finally, we prove that a residually finite group whose finite quotients admit inverse ambiguous automorphisms is abelian. However, we prove the existence of infinite residually finite abelian groups whose finite quotients all admit such automorphisms, but the group itself does not.