Hypermaps over non-abelian simple groups and strongly symmetric generating sets (Q6133154)

Let \(G\) be a \(2\)-generator group. The authors define a generating pair \( (x,y)\) for \(G\ \)to be symmetric if the mapping \((x,y)\longmapsto (x^{-1},y^{-1})\) can be extended to an automorphism of \(G\) and \(G\) is strongly symmetric if every generating pair is symmetric. It is known that a finite \(2\)-generator group is strongly symmetric if and only if every orientably regular hypermap with monodromy group \(G\) is reflexible [\textit{N. Gill} et al., Cherlin's conjecture for finite primitive binary permutation groups. Cham: Springer (2022; Zbl 1515.20011), Lemma 7]. This note proves the following theorem: A finite non-abelian simple group \(G\) is strongly symmetric if and only if \(G\cong \mathrm{PSL}(2,q)\) for some prime power \(q\).