Investigating transversals as generating sets for groups

TL;DR

This paper extends previous results on the ability to find generating transversals in groups of rank up to 4 and under certain conditions for higher ranks, broadening understanding of group generation properties.

Contribution

It generalizes the existence of generating transversals for groups of rank up to 4 and introduces conditions for higher ranks, including divisibility and malnormality.

Findings

Extended the result to groups of rank 4.

Provided conditions for arbitrary rank groups based on divisors.

Established results for malnormal subgroups in arbitrary rank groups.

Abstract

In [3] is was shown that for any group $G$ whose rank (i.e., minimal number of generators) is at most 3, and any finite index subgroup $H\leq G$ with index $[G:H]\geq rank(G)$, one can always find a left-right transversal of $H$ which generates $G$. In this paper we extend this result to groups of rank at most 4. We also extend this to groups $G$ of arbitrary (finite) rank $r$ provided all the non-trivial divisors of $[G:core_G(H)]$ are at least $2r-1$. Finally, we extend this to groups $G$ of arbitrary (finite) rank provided $H$ is malnormal in $G$.