Hypermaps Over Non-Abelian Simple Groups and Strongly Symmetric Generating Sets

A generating pair x, y for a group G is said to be symmetric if there exists an automorphism ϕx,y of G inverting both x and y, that is, xϕx,y = x−1 and yϕx,y = y−1. Similarly, a group G is said to be 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. Mathematics Subject Classifications: 05C10, 05C25, 20B25.