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

A generating pair $x, y$ for a group $G$ is said to be \textbf{\textit{symmetric}} if there exists an automorphism $φ_{x,y}$ of $G$ inverting both $x$ and $y$, that is, $x^{φ_{x,y}}=x^{-1}$ and $y^{φ_{x,y}}=y^{-1}$. Similarly, a group $G$ is said to be \textbf{\textit{strongly symmetric}} if $G$ can be generated with two elements and if all generating pairs of $G$ are symmetric. In this paper we classify the finite strongly symmetric non-abelian simple groups. Combinatorially, these are the finite non-abelian simple groups $G$ such that every orientably regular hypermap with monodromy group $G$ is reflexible.

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