Strong map-symmetry of SL(3, K) and PSL(3, K) for every finite field K

Abstract

In this paper, we show that for any finite field K, any pair of map-generators (that is when one of the generators is an involution) of SL(3,K) and PSL(3,K) has a group automorphism that inverts both generators. In the theory of maps, this corresponds to say that any regular oriented map with automorphism group SL(3,K) or PSL(3,K) is reflexible, or equivalently, there are no chiral regular maps with automorphism group SL(3,K) or PSL(3,K). As remarked by Leemans and Liebeck, also SU(3,K) and PSU(3,K) are not automorphism groups of chiral regular maps. These two results complete the work of the above authors on simples groups supporting chiral regular maps.