Inverse ambiguous functions on some finite non-abelian groups
Let \(G\) be a group and \(N(G) =\{x \in G | x \neq x^{-1}\}\). An inverse ambiguous function on \(G\) is a bijective function \(f : G \to G\) such that \(f^{-1}(x) = (f(x))^{-1}\) for all \(x \in G\). \textit{D. J. Schmitz} [Aequationes Math. 91, No. 2, 373--389 (2017; Zbl 1372.39027)] showed that there exists an inverse ambiguous function on \(G\) if and only if \(|N(G)|\) is a multiple of \(4\). In this paper the authors employ this criterion to determine when inverse ambiguous functions exist on some common nonabelian groups, specifically the symmetric groups, the alternating groups, and the general linear groups of order \(2\) over a finite field. To count the number of elements having order greater than \(2\), they employ elementary combinatorial and group methods.
scientific article; zbMATH DE number 6944068