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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
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