  Let $P_k$ be the subgroup generated by $k$th powers of primitive elements in
$F_r$, the free group of rank $r$. We show that $F_2/P_k$ is finite if and only
if $k$ is $1$, $2$, or $3$. We also fully characterize $F_2/P_k$ for $k =
2,3,4$. In particular, we give a faithful nine dimensional representation of
$F_2/P_4$ with infinite image.
