Finite-index subgroup quasi-isometry

If $H$ is a finite-index subgroup of a finitely generated group $G$, then $H$ is finitely generated and its inclusion into $G$ is a quasi-isometry. Finite generation follows from Schreier's lemma, while finitely many coset representatives give coarse surjectivity and uniformly bounded distortion.

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.