1) Let $F$ be the free group on the three generators $x,y,z$. For non-zero integers $r,s,t$
then CLAIM: the subgroup of $F$ generated by $x^r , y^s , z^t$ is freely generated by these elements.
2) Let $H$ be the subgroup of $F(\{x,y\})$ generated by $x^2 , y^2 , xy , yx$
then CLAIM: $H$ is not freely generated by these elements.
Can anyone help me?