Transversals as generating sets in finitely generated groups

Abstract

We explore transversals of finite index subgroups of finitely generated groups. We show that when $H$ is a subgroup of a rank $n$ group $G$ and $H$ has index at least $n$ in $G$ then we can construct a left transversal for $H$ which contains a generating set of size $n$ for $G$, and that the construction is algorithmic when $G$ is finitely presented. We also show that, in the case where $G$ has rank $n\le 3$, there is a simultaneous left-right transversal for $H$ which contains a generating set of size $n$ for $G$. We finish by showing that if $H$ is a subgroup of a rank $n$ group $G$ with index less than $3\cdot 2^{n-1}$, and $H$ contains no primitive elements of $G$, then $H$ is normal in $G$ and $G/H\cong C_2^n$.

15 pages, 6 figures. This is the version submitted for publication

References in corpus (2)

• Measure Preserving Words are Primitive
• Coset intersection graphs for groups

Cited by in corpus (1)

• Investigating transversals as generating sets for groups