Let $G$ be a group with a free subgroup of rank $2$. Let $H\leq G$ be such that $[G:H]<\infty$. Then $H$ also contains a free subgroup of rank $2$.

Question

I am having difficulties in solving the following problem.

Let $G$ be a group with a free subgroup of rank $2$. Let $H\leq G$ be such that $[G:H]<\infty$. Then $H$ also contains a free subgroup of rank $2$.

We know by Nielsen-Schreier theorem that a subgroup of a free group is also free. But in this problem $G$ is not necessarily free but contains a free subgroup. How to approach this problem? Any hint or idea will be highly appreciated. Thanks in anticipation.

Answer 1

Let $F$ be the free subgroup of rank $2$ in $G$.

Then $|G:H|$ finite implies that $k := |F:H \cap F|$ is also finite, and by the Nielsen-Schreier Theorem $H \cap F$ is free of rank $k+1$.

So $H \cap F$ and hence also $H$ contains a free subgroup of rank $2$.

Answer 2

Let $n=|G:H|$ be the index of $H$ in $G$, and let $a, b\in G$ be the generators of $F$. Then $\langle a^{n!}, b^{n!}\rangle$ is free of rank two, and is contained in $H$.