Every Polish group has a non-trivial topological group automorphism

TL;DR

This paper proves that all Polish groups with more than two elements have non-trivial automorphisms, impacting the understanding of homogeneous spaces and their group structures.

Contribution

It establishes that every non-trivial Polish group admits a non-trivial automorphism, showing a fundamental property of Polish groups.

Findings

Every Polish group with more than two elements has a non-trivial automorphism.

A hypothetical uniquely homogeneous Polish space with more than two points cannot be a semitopological group.

Abstract

We prove that every Polish group with more than two elements admits a non-trivial topological group automorphism. As a consequence, a hypothetical uniquely homogeneous Polish space with more than two points cannot be a semitopological group.