A consequence of the Nielsen-Schreier theorem is the following: If a group generated by $n$ elements, then every subgroup of finite index $k$ is generated by $kn−k+1$ elements. See also this aops discussion; there jmerry gives a direct algebraic proof.
A consequence of the Nielsen-Schreier theorem is the following: If a group generated by $n$ elements, then every subgroup of finite index $k$ is generated by $kn−k+1$ elements. See also this aops discussion.