Nielsen–Schreier Theorem

Nielsen–Schreier Theorem. Let $F$ be a free group and let $H \le F$ be a subgroup . Then $H$ is a free group.

If $F$ has finite rank $n$ and the index $[F:H]$ is finite, then $H$ has finite rank and satisfies the Schreier index formula

This theorem is proved by constructing an explicit free generating set for $H$ from a transversal of cosets; Schreier's lemma provides the standard generating set used in the proof. It is a foundational result in combinatorial group theory.

Proof sketch. Choose a Schreier transversal $T$ for the cosets of $H$ in $F$. Schreier’s method produces generators of $H$ of the form $t s (\overline{ts})^{-1}$ (with $t\in T$ and $s$ in a free generating set of $F$), and one shows these generators satisfy no relations, hence freely generate $H$.