# Proving Neilsen-Schreier Theorem using "only free groups act freely on a tree"

This is exercise II.9.16 from Aluffi's Algebra: Chapter 0.

Before tackling this theorem, I have proved (rather loosely because I don't know much about graph theory) that the Cayley diagram of a free group is free, and that groups acts freely on a tree iff it is a free group. According to the book, with these in mind, we are ready to prove the theorem.

The very first idea that I thought of is to prove that subgroups of a free group act freely on a tree. How can this be done with minimum use of knowledge of graph theory?

Any hints, solutions or reference of other sources is appreciated.

• Have a look at Serre's, "Trees". Feb 3, 2020 at 22:20

You already know that a group is free iff it acts freely on a tree. Let $$G$$ be a free group and $$H$$ be a subgroup of $$G$$. Then there is a tree on which $$G$$ acts freely. This action induces an action of $$H$$ on the same tree. It is easy to check that this action is also free. Since $$H$$ acts freely on a tree, $$H$$ is free.