A note on invariant transversals for normal subgroups

TL;DR

This paper investigates the existence of invariant transversals for normal subgroups in groups, providing counterexamples to a conjecture when the subgroup is abelian and the group is finite.

Contribution

It offers new insights into the conditions under which invariant transversals exist, including counterexamples that challenge previous conjectures.

Findings

Counterexamples for abelian normal subgroups in finite groups

Invariance of transversals does not always hold in these cases

Provides a deeper understanding of subgroup transversals in group theory

Abstract

The existence of invariant transversals for a normal subgroup $H$ in a group $G$ is investigated. This yields counterexamples to a conjecture in case $H$ is abelian and $G$ is finite.