Combinatorial categories and permutation groups

Abstract

The regular objects in various categories, such as maps, hypermaps or covering spaces, can be identified with the normal subgroups N of a given group Γ , with automorphism group isomorphic to Γ  / N. It is shown how to enumerate such objects with a given finite automorphism group G, how to represent them all as quotients of a single regular object U(G), and how the outer automorphism group of Γ  acts on them. Examples constructed include kaleidoscopic maps with trinity symmetry.