Lifting Generators in Connected Lie Groups

TL;DR

This paper investigates when generating sets in connected Lie groups can be lifted through epimorphisms, revealing that the problem reduces to abelian cases and establishing conditions related to the Gaschütz lemma and group perfection.

Contribution

It demonstrates that lifting generators in connected Lie groups depends on abelianization, proves the Gaschütz lemma for perfect groups, and introduces the Gaschütz rank to measure failure in non-perfect groups.

Findings

Generators can be lifted if and only if they lift in the abelianization map.

Connected perfect Lie groups satisfy the Gaschütz lemma.

The Gaschütz rank quantifies how non-perfect groups fail to satisfy the lemma.

Abstract

Given an epimorphism between topological groups $f:G\to H$, when can a generating set of $H$ be lifted to a generating set of $G$? We show that for connected Lie groups the problem is fundamentally abelian: generators can be lifted if and only if they can be lifted in the induced map between the abelianisations (assuming the number of generators is at least the minimal number of generators of $G$). As a consequence, we deduce that connected perfect Lie groups satisfy the Gasch\"utz lemma: generating sets of quotients can always be lifted. If the Lie group is not perfect, this may fail. The extent to which a group fails to satisfy the Gasch\"utz lemma is measured by its \emph{Gasch\"utz rank}, which we bound for all connected Lie groups, and compute exactly in most cases. Additionally, we compute the maximal size of an irredundant generating set of connected abelian Lie groups, and discuss connections between such generation problems with the Wiegold conjecture.