Some examples of invariably generated groups

TL;DR

This paper constructs examples of finitely invariably generated groups with specific subgroup properties, demonstrating that the properties of being IG or FIG are not preserved under finite index subgroups, and provides the first examples of finitely generated IG groups that are not FIG.

Contribution

It provides the first examples of FIG groups with subgroups that are not IG and finitely generated IG groups that are not FIG, addressing open questions in the field.

Findings

Existence of a FIG group with a subgroup that is not IG.

Existence of finitely generated IG groups that are not FIG.

Answers to open questions by Wiegold, Kantor, Lubotzky, Shalev, and Cox.

Abstract

A group $G$ is invariably generated (IG) if there is a subset $S \subseteq G$ such that for every subset $S' \subseteq G$, obtained from $S$ by replacing each element with a conjugate, $S'$ generates $G$. $G$ is finitely invariably generated (FIG) if, in addition, one can choose such a subset $S$ to be finite. In this note we construct a FIG group $G$ with an index $2$ subgroup $N \lhd G$ such that $N$ is not IG. This shows that neither property IG nor FIG is stable under passing to subgroups of finite index, answering questions of Wiegold and Kantor, Lubotzky, Shalev. We also produce the first examples of finitely generated IG groups that are not FIG, answering a question of Cox.