Combinatorial categories and permutation groups

TL;DR

This paper explores the classification and enumeration of regular objects like maps and hypermaps through group-theoretic methods, introducing a unified approach to represent and analyze their automorphism groups.

Contribution

It presents a novel framework for representing all regular objects with a given automorphism group as quotients of a universal object, and examines their symmetries via outer automorphisms.

Findings

Enumeration method for objects with a specified automorphism group

Construction of a universal regular object U(G)

Examples include kaleidoscopic maps with trinity symmetry

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 \Gamma, with quotient group isomorphic to \Gamma/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 they are acted on by the outer automorphism group of \Gamma. Examples constructed include kaleidoscopic maps with trinity symmetry.