Intuitive understanding of the Reidemeister-Schreier Theorem
Solution 1:
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.
- Find a 2-complex with fundamental group $G$.
- Find the covering space corresponding to $H$.
- Lift the structure of the 2-complex to the covering space.
- 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.