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 $G$, we can construct a left transversal for $H$ which contains a generating set of size $n$ for $G$; this construction is algorithmic when $G$ is finitely presented. We also show that, in the case where $G$ has rank $n\leq 3$, 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 $G/H\cong C_{2}^{n}$.

Theorem 1 asserts that \emph{in a finitely generated prosoluble group, every subgroup of finite index is open}. This generalises an old result of Serre about pro-$p$ groups. It follows by a standard argument from Theorem 2: \emph{in a $d$-generator finite soluble group, every element of the derived group is equal to a product of $72d^2 +46d$ commutators}. This result about finite soluble groups is proved by induction on the order of the group, and is elementary though rather intricate. The essence of the proof lies in reducing the problem to one about the number of solutions of quadratic equations over a finite field. Corollaries include the following. \emph{Let $\Gamma$ be a finitely generated prosoluble group. Then each term of the lower central series of $\Gamma$ and each power subgroup $\Gamma ^n$ is closed}. 1991 Mathematics Subject Classification: 20E18, 20D10.