Realisation of groups as automorphism groups in categories

Gareth A. Jones

Published 2018-07-02Version 1

It is shown that in various categories, including certain categories consisting of maps or hypermaps, oriented or unoriented, and their coverings, every countable group $A$ is isomorphic to the automorphism group of some object, which can be chosen to be finite if $A$ is finite. In particular, every finite group is isomorphic to the automorphism group of a dessin d'enfant of any given hyperbolic type. The method of proof, involving maximal subgroups of various triangle groups, yields a simple construction of a regular map whose automorphism group contains an isomorphic copy of every finite group.