  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 $n \leq3$, 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 $G/H \cong C_{2}^{n}$.

Group Theory