Nilpotent primer hypermaps with hypervertices of valency a prime
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Nilpotent primer hypermaps with hypervertices of valency a prime Shaofei Du1 · Kai Yuan1 Received: 18 May 2018 / Accepted: 12 September 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract A (face-)primer hypermap is a regular oriented hypermap whose group G of the orientation-preserving automorphisms acts faithfully on its hyperfaces. In this paper, we investigate the primer hypermaps H such that G is nilpotent and the hypervertex valency is a prime p (simply call them P N p hypermaps). We prove that G is a finite pgroup and the number of hyperfaces of any P N p hypermap is bounded by a function of the nilpotent class of G. Moreover, we show that for any integer c 1, there is a unique P N p hypermap Hc of class c attaining the bound and every other P N p hypermap H of the class at most c is a quotient of Hc , see Theorem 1.1. The main theorem of this paper generalizes the result of Conder et al. (Algebraic Comb 4:863–874, 2016). Keywords Regular map · Hypermap · Nilpotent group Mathematics Subject Classification Primary 05C10 · 05C25; Secondly 20B25
1 Introduction Topologically, a hypermap is a 2-cell embedding of a connected bipartite graph G (may have multiple edges) into a compact and connected surface S without border, where the vertices of G in two biparts are, respectively, called the hypervertices and hyperedges of the hypermap, and the connected regions of S\G are called hyperfaces. A hypermap H is orientable if the underlying surface S is orientable. For an orientable hypermap H, the group Aut (H) of orientation-preserving automorphisms acts freely on its darts D (the edges of G). Furthermore, when Aut (H) acts regularly on D, we
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Shaofei Du [email protected] Kai Yuan [email protected]
1
School of Mathematical Sciences, Capital Normal University, Beijing 100048, China
123
Journal of Algebraic Combinatorics
call H is regular. In other wards, a regular hypermap has the largest possible number of symmetries. From now on, all hypermaps are regular oriented hypermaps unless otherwise specified. As usual, every oriented hypermap can be represented by an ordered triple H = (G; a, b) (see Sect. 2). Identify G with its right regular representation R(G) = Aut (H), and denote the set of hyperfaces by F. Then, the stabilizer of G for a hyperface is H = a. If G induces a faithful action on F, then we call H (face-) primer. For any hypermap (G; a, b), the quotient hypermap (G/HG ; a, b) induced by the core HG of H in G is called the primer hypermap of H. Any regular hypermap covers its primer hypermap, see [2]. Generally speaking, classifying regular hypermaps is very complicated. The first step might be to determine the primer hypermaps. Based on the knowledge of primer hypermaps, one may determine general hypermaps. Moreover, one may see that to recover a hypermap from its primer hypermap is essentially an extension problem of a group by a cyclic group. Classifying regular hypermaps and maps with given underlying graph, surface and automorphism group are central problems
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