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 graph, and the Cayley graph of G is a 2|X|-regular
tree, which we will call 𝒯.
If H is any subgroup 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 characteristic of 𝒢. Since H≅π1(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:
rank(H)=n(rank(G)-1)+1. |
Title | proof of Nielsen-Schreier theorem and Schreier index formula |
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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 |