This question comes from Rolfsen's Knots and Links.
I would like to figure out why the free product of the trefoil group and $\mathbb{Z}$ is not free.
There are a couple of ways I am trying to work in. For instance, if we add the relation $a^2 = 1$ (rigorously the normal closure of $a^2$) to the the trefoil group $\langle a, b\mid a^2 = b^3\rangle$, then $\mathbb{Z}\ast\langle a, b\mid a^2 = b^3\rangle$ turns to be the free product $G = \mathbb{Z}\ast\mathbb{Z}_2\ast\mathbb{Z}_3$. I would like to show that adding one relation to a free product will not give a group isomorphic to $G$, but I don't know how.
Any ideas will be greatly appreciated!