proof of Nielsen-Schreier theorem and Schreier index formula

While there are purely algebraic proofs of the Nielsen-Schreier theorem, a much easier proof is available through geometric group theory.

Let G be a group which is free on a set X. Any group acts freely on its Cayley graphMathworldPlanetmath, and the Cayley graph of G is a 2|X|-regularPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath tree, which we will call 𝒯.

If H is any subgroupMathworldPlanetmathPlanetmath of G, then H also acts freely on 𝒯 by restriction. Since groups that act freely on trees are free, H is free.

Moreover, we can obtain the rank of H (the size of the set on which it is free). If 𝒢 is a finite graph, then π1(𝒢) is free of rank -χ(𝒢)-1, where χ(𝒢) denotes the Euler characteristicMathworldPlanetmath of 𝒢. Since Hπ1(H\𝒯), the rank of H is χ(H\𝒯). If H is of finite index n in G, then H\𝒯 is finite, and χ(H\𝒯)=nχ(G\𝒯). Of course -χ(G\𝒯)+1 is the rank of G. Substituting, we obtain the Schreier index formula:

Title proof of Nielsen-Schreier theorem and Schreier index formula
Canonical name ProofOfNielsenSchreierTheoremAndSchreierIndexFormula
Date of creation 2013-03-22 13:56:02
Last modified on 2013-03-22 13:56:02
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 7
Author mathcam (2727)
Entry type Proof
Classification msc 20E05
Classification msc 20F65
Related topic ScheierIndexFormula