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Homeproof of Nielsen-Schreier theorem and Schreier index formula

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# 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 $\mathcal{T}$.

If $H$ is any subgroup of $G$, then $H$ also acts freely on $\mathcal{T}$ 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 $\mathcal{G}$ is a finite graph, then $\pi_{1}(\mathcal{G})$ is free of rank $-\chi(\mathcal{G})-1$, where $\chi(\mathcal{G})$ denotes the Euler characteristic of $\mathcal{G}$. Since $H\cong\pi_{1}(H\backslash\mathcal{T})$, the rank of $H$ is $\chi(H\backslash\mathcal{T})$. If $H$ is of finite index $n$ in $G$, then $H\backslash\mathcal{T}$ is finite, and $\chi(H\backslash\mathcal{T})=n\chi(G\backslash\mathcal{T})$. Of course $-\chi(G\backslash\mathcal{T})+1$ is the rank of $G$. Substituting, we obtain the Schreier index formula:

$\mathrm{rank}(H)=n(\mathrm{rank}(G)-1)+1.$ |

## Mathematics Subject Classification

20E05*no label found*20F65

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## Comments

## Weird linking issue

After a couple of PMs undoubtedly thinking the other was an idiot, mathprof and I have stumbled on to a weird bug -- The word "graph" in this entry links to the correct phrase in page images mode, but a different entry in html mode. Or is this a "feature"? Is there an easy fix?

Cam

## Re: Weird linking issue

I'm not sure how easy of a fix this is, but a fix does exist. Namely, you can replace each occurence of "graph" that links to where you do not want it to by \PMlinkname{graph}{Graph}. I would not recommend doing this in places like in the phrase "Cayley graph", which seems to link to where you would want it to in html mode. This is the best that I can do.

Warren

## Re: Weird linking issue

Actually, a better idea (in my humble opinion) would be to replace the phrase "finite graph" with \PMlinkname{finite graph}{Graph}.

Warren