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 be a group which is free on a set . Any group acts freely on its Cayley graph, and the Cayley graph of is a -regular tree, which we will call .
If is any subgroup of , then also acts freely on by restriction. Since groups that act freely on trees are free, is free.
Moreover, we can obtain the rank of (the size of the set on which it is free). If is a finite graph, then is free of rank , where denotes the Euler characteristic of . Since , the rank of is . If is of finite index in , then is finite, and . Of course is the rank of . Substituting, we obtain the Schreier index formula:
|Title||proof of Nielsen-Schreier theorem and Schreier index formula|
|Date of creation||2013-03-22 13:56:02|
|Last modified on||2013-03-22 13:56:02|
|Last modified by||mathcam (2727)|