# Schreier index formula

Let $F$ be a free group of finite rank, and let $H$ be a subgroup (http://planetmath.org/Subgroup) of finite index in $F$. By the Nielsen-Schreier theorem, $H$ is free. The Schreier index formula states that

 $\operatorname{rank}(H)=|F:H|\cdot(\operatorname{rank}(F)-1)+1.$

This implies more generally that if $G$ is a group generated by $m$ elements, then any subgroup of index $n$ in $G$ can be generated by at most $nm-n+1$ elements.

Title Schreier index formula SchreierIndexFormula 2013-03-22 13:56:18 2013-03-22 13:56:18 yark (2760) yark (2760) 14 yark (2760) Theorem msc 20E05 ProofOfNielsenSchreierTheoremAndSchreierIndexFormula