Schreier's Lemma - Knowlpedia

Schreier's lemma. Let $G$ be a group generated by $S$ , let $H\le G$ be a subgroup, and choose a right transversal $T$ for $H$ in $G$ . For $g\in G$ , write $\overline g\in T$ for the unique representative of the right coset $Hg$ .

Then $H$ is generated by

t\,u\,(\overline{tu})^{-1} \qquad (t\in T,\ u\in S\cup S^{-1}).

If $S=S^{-1}$ , it suffices to let $u$ range over $S$ .

Remarks

This is a core tool for proving results about generators of subgroups, including the Nielsen–Schreier theorem.