Schreier's lemma

If a group $G$ is generated by $S$, a subgroup $H$ has a set $T$ of right-coset representatives, and $\overline{ts}$ denotes the representative of $Hts$, then $H$ is generated by the Schreier generatorsIn particular, every finite-index subgroup of a finitely generated group is finitely generated.

Given a presentation of a group and a Schreier transversal for a subgroup, the Reidemeister–Schreier theorem gives a presentation of the subgroup. Its generators are the Schreier generators $ts{\overline{ts}}^{-1}$, and its relators are rewritten conjugates of the original relators.

For a subgroup $H$ of a group generated by a set $S$, choose representatives $t$ of right cosets $Ht$. If $\overline{tg}$ represents $Htg$, then $tg{\overline{tg}}^{-1}$ lies in $H$. These Schreier generators generate $H$: inserting successive coset representatives along a word telescopes the word into such factors. Lifted relators can be rewritten in the same factors.