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

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

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
Canonical name	SchreierIndexFormula
Date of creation	2013-03-22 13:56:18
Last modified on	2013-03-22 13:56:18
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	14
Author	yark (2760)
Entry type	Theorem
Classification	msc 20E05
Related topic	ProofOfNielsenSchreierTheoremAndSchreierIndexFormula