Regular maps with almost Sylow-cyclic automorphism groups, and classification of regular maps with Euler characteristic −p2

A regular map M is a cellular decomposition of a surface such that its automorphism group Aut(M) acts transitively on the flags of M. It can be shown that if a Sylow subgroup P⩽Aut(M) has order coprime to the Euler characteristic of the supporting surface, then P is cyclic or dihedral. This observation motivates the topic of the current paper, where we study regular maps whose automorphism groups have the property that all their Sylow subgroups contain a cyclic subgroup of index at most 2. The main result of the paper is a complete classification of such maps. As an application, we show that no regular maps of Euler characteristic −p2 exist for p a prime greater than 7.