# zbMATH — the first resource for mathematics

Classification of regular maps whose automorphism groups are 2-groups of maximal class. (English) Zbl 1271.05103
The authors determine all orientably-regular maps whose group of orientation-preserving automorphisms is a $$2$$-group of maximal class – that is, a group of order $$2^n$$ and nilpotency class $$n-1$$ (although the authors do not explain this and give an inappropriate reference for it). Every $$2$$-group of maximal class is either dihedral, semi-dihedral or generalised quaternion; see J. A. de Séguier’s book [Théorie des groupes finis. Éléments de la théorie des groupes abstraits. Paris: Gauthier-Villars (1904; JFM 36.0187.02)], or D. Gorenstein’s [Finite Groups. New York-Evanston-London: Harper and Row (1968; Zbl 0185.05701)]. The group of orientation-preserving automorphisms of every orientably-regular map is generated by an involution and one other element. The involution cannot be central when the group is non-abelian, so the generalised quaternion case is easily eliminated. In the other two cases (where $$G$$ is dihedral or semi-dihedral), the map must be a regular embedding of a cycle or multi-cycle, or its dual (a dipole).
##### MSC:
 05E18 Group actions on combinatorial structures 20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures 20D15 Finite nilpotent groups, $$p$$-groups 05C10 Planar graphs; geometric and topological aspects of graph theory 57M15 Relations of low-dimensional topology with graph theory 57M60 Group actions on manifolds and cell complexes in low dimensions
##### Keywords:
regular maps; finite 2-groups; automorphism groups
Full Text: