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

Conder, Marston; Potočnik, Primož and Širáň, Jozef (2010). Regular maps with almost Sylow-cyclic automorphism groups, and classification of regular maps with Euler characteristic −p². Journal of Algebra, 324(10) pp. 2620–2635.

DOI: https://doi.org/10.1016/j.jalgebra.2010.07.047

Abstract

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.

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