Normal generation of locally compact groups

TL;DR

This paper addresses a long-standing open problem in group theory by proving that certain topological groups can be normally generated by a single element, under specific conditions excluding infinite discrete quotients.

Contribution

It provides a positive answer to the topological analogue of the 1970s open problem regarding normal generation of finitely generated perfect groups.

Findings

Topological groups can be normally generated by a single element under certain conditions.

Excluding infinite discrete quotients is likely necessary for the result.

The problem remains open in the purely algebraic setting.

Abstract

It is a well-known open problem since the 1970s whether a finitely generated perfect group can be normally generated by a single element or not. We prove that the topological version of this problem has an affirmative answer as long as we exclude infinite discrete quotients (which is probably a necessary restriction).