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

TL;DR

This paper classifies finite non-abelian simple groups where every generating pair is symmetric, linking group properties to the combinatorial symmetry of associated hypermaps.

Contribution

It provides a complete classification of strongly symmetric non-abelian simple groups, connecting algebraic group properties with hypermap symmetry.

Findings

Identifies all finite non-abelian simple groups with symmetric generating pairs.

Establishes that these groups correspond to hypermaps that are always reflexible.

Links group automorphisms to hypermap symmetries.

Abstract

A generating pair $x, y$ for a group $G$ is said to be \textbf{\textit{symmetric}} if there exists an automorphism $\varphi_{x,y}$ of $G$ inverting both $x$ and $y$, that is, $x^{\varphi_{x,y}}=x^{-1}$ and $y^{\varphi_{x,y}}=y^{-1}$. 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.