Inverse ambiguous functions on some finite non-abelian groups

Abstract

In Schmitz (Aequ Math 91:373–389, 2017), the first author defines an “inverse ambiguous function” on a group G to be a bijective function $f:G \rightarrow G$ satisfying the functional equation $f^{-1}(x) = (f(x))^{-1}$ for all $x \in G$. Using a simple criterion involving the number of elements in G not equal to their own inverse, the classification of finite abelian groups admitting inverse ambiguous functions is achieved. In this paper we aim to extend the results from (2017) to determine the existence of inverse ambiguous functions on members of certain families of non-abelian groups, namely the symmetric groups $S_n$, the alternating groups $A_n$, and the general linear groups GL(2, q) over a finite field $\mathbb {F}_q$.