Classification of regular maps of prime characteristic revisited: Avoiding the Gorenstein-Walter theorem

Abstract

Breda, Nedela and Širáň (2005) classified the regular maps on surfaces of Euler characteristic −p for every prime p. This classification relies on three key theorems, each proved using the highly non-trivial characterisation of finite groups with dihedral Sylow 2-subgroups, due to D. Gorenstein and J.H. Walter (1965). Here we give new proofs of those three facts (and hence the entire classification) using somewhat more elementary group theory, using without referring to the Gorenstein-Walter theorem.