Schreier's Lemma

Schreier’s Lemma: Let $G$ be a group generated by a set $S$ (i.e. every element of $G$ is a finite product of elements of $S\cup S^{-1}$). Let $H\le G$ be a subgroup . Choose a right transversal $T\subseteq G$ for $H$ in $G$, meaning: for every right coset $Hg$ there exists a unique $t\in T$ with $Hg=Ht$.

For $t\in T$ and $s\in S$, let $\overline{ts}\in T$ be the unique representative with $Hts=H\overline{ts}$. Then $H$ is generated by the Schreier generators $t\,s\,(\overline{ts})^{-1}\in H \qquad (t\in T,\ s\in S).$

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

Proof sketch: Any $h\in H$ is a word in $S\cup S^{-1}$. Track the successive coset representatives in $T$ as you read the word; inserting and removing these representatives rewrites $h$ as a product of elements of the form $t s (\overline{ts})^{-1}$, showing these elements generate $H$.