Note on Mapping Class Groups of Finite Spaces

TL;DR

This paper studies the mapping class groups of finite and certain non-Hausdorff spaces, revealing their structure and showing every finite group can be realized as such a group.

Contribution

It establishes an isomorphism between the mapping class group of a finite space and the homeomorphism group of its $T_0$ quotient, and demonstrates that every finite group arises as a mapping class group.

Findings

Mapping class group of a finite space is isomorphic to the homeomorphism group of its $T_0$ quotient.

Every finite group can be realized as the mapping class group of some finite, $T_0$ space.

Abstract

We investigate the mapping class groups of a class of non-Hausdorff topological spaces which includes finite spaces. We show that the mapping class group of a finite space is isomorphic to the homeomorphism group of its $T_0$ quotient. As a corollary, we show that every finite group is the mapping class group of some finite, $T_0$ space.