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

\operatorname{rank}(H) = 1 + [F:H]\,(n-1).

Remarks

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.