Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 143 / 1 / c / Solution

Let $r=d(G)$ be the rank of a group, let $S$ be a generating set of size $r$, and suppose $[G:H]=d$. Choose a Schreier transversal adapted to a spanning tree in the Schreier coset graph. By Schreier's lemma, $H$ is generated by the elementsThere are $dr$ candidates. The $d-1$ oriented edges in the spanning tree give trivial candidates, leaving at most $dr-(d-1)$. This proves the Schreier index-rank inequality