I was wondering whether there is some injective homomorphism from $\mathbb{Z}\star\mathbb{Z}$ to $\mathbb{Z}\times\mathbb{Z}$, where with $\star$ I have denoted the free product, and with $\times$ the direct sum?
My guess is that there is no injective homomorphism, since if there were such a homomorphism then it would suffice to define it on the generators of $\mathbb{Z}\star\mathbb{Z}$ where these would be sent to the generators of $\mathbb{Z}\times\mathbb{Z}$, but cannot quite see how to argue?