T/F question on free groups

Question

Is this statement True or False-

If F is a free group with basis {$x,y$} and H is the subgroup generated by {$x^2,y^2,xy,yx$} then H is a free group of rank $3$.

What should be my approach to solve it. I do not know how to proceed.

Answer 1

Any subgroup of a free group is free. To compute the rank of $H$, note that $H$ is the kernel of the map $F \to \mathbb{Z}_2$ given by $x \to 1, y \to 1$, and use the Nielsen-Schreier theorem. Alternatively, show that the abelianization of $H$ is isomorphic to $\mathbb{Z}^3$ by considering it as the kernel above (and note that $yx = y^2(xy)^{-1}x^2$).