Möbius inversion of some classical groups and their applications to the enumeration of regular hypermaps

Abstract

For any group Γ abstractly defined by some finite presentation, a method (called Hall's method) is known for calculating the number dΓ(G) of normal subgroups N of Γ for which the quotient group Γ/N is of some preset isomorphism type G. This technique is dependent on the knowledge of the Móius function μ_G on the subgroup lattice of G. We apply this method by use of the following proposition: if Γ is any group with a finite 2-generator presentation and with relator set R, then d_{Γ(G) gives the number of regular oriented hypermaps with automorphism group G which satisfy certain well-defined local properties determined by the relators R. In chapter 1 we review the established theories of Hall's method and of regular hypermaps (including unoriented hypermaps) and discuss their relationship as above. In chapter 2 the function} μ_G is calculated for G = PSL₂(q) or PGL₂(q) for any prime power q (extending an existing result for primes). In chapter 3 these results are applied to explicitly make some enumerations of various specific categories of regular hypermaps. (However some other enumerations are made by a different method, based on trace). Chapters 4 and 5 specialise mostly to triangular maps. Chapter 4 examines the local properties of regular oriented triangular maps with automorphism group G ˜= PSL₂(q) or PGL₂(q) for some q, in particular how to distinguish two such maps with the same automorphism group. Chapter 5 describes how some of these same maps may be constructed in a different way, each one as the unique triangular imbedding of a graph with vertices defined as the elements of a particular conjugacy class in G. (D82616)

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