# Intuitive understanding of the Reidemeister-Schreier Theorem

I am reading Combinatorial Group Theory by Lyndon and Schupp, and I'm having some trouble getting through the proof of the Reidemeister-Schreier theorem (page 103 in that book) - you can read that part on google books. I'm wondering if someone can give me some intuition for this result, which might help me with the path of the proof. You don't need to be very precise; it just seems that given a presentation $\langle X=\{x_1,...,x_n\};R=\{r_1,...,r_m\}\rangle$ of a group $G=F/N$ (with $F$ free and $N$ the closure of $R$) and a set $T$ (a Schreier transversal) of elements $t_i$ such that as sets

$$(Ht_1)\sqcup (Ht_2)\sqcup\dots\sqcup(Ht_n)=F$$

for $H$ subgroup of $G$, the process of constructing a presentation $\langle X',R' \,\rangle$ of $H$ should be easy to explain intuitively.

• Note that in your displayed formula, you want the union to be a disjoint union, i.e., the cosets to be distinct. – Andreas Caranti May 21 '13 at 17:07
• I'm not sure I understand what's the problenm here: we are given $\,H\le G=F/N\,$ , so what construction process do you want?? – DonAntonio May 21 '13 at 19:22
• The Reidemeister-Schreier theorem tells you how to get a presentation of $H$ given a presentation of $G$. I have edited a little to make that more clear. – levitopher May 21 '13 at 21:34
• you should mimic as in the work of journals.cambridge.org/action/… – janmarqz Jun 4 '13 at 19:44

## 1 Answer

Perhaps the most intuitive explanation of Reidemeister-Schreier -especially if you are comfortable with topology- is using covering spaces. Basically, Reidemeister-Schreier can be thought of as follows.

1. Find a 2-complex with fundamental group $G$.
2. Find the covering space corresponding to $H$.
3. Lift the structure of the 2-complex to the covering space.
4. Compute fundamental group.

A more complete explanation can be found in an answer of Qiaochu Yuan here, while I wrote out a rather complicated worked example (without pictures!) in an answer here.

However, this gives you a presentation for your group rather than, for example, the actual generators of the kernel of a map.