I have come across this question in a course about Cayley graphs recently. I don't really have a clue how to answer this question. Here it is.
Consider two groups $A,B$. Let $K< A*B$ be the kernel of the homomorphism $\phi: A * B \to A \times B$ that extends the inclusions $A \hookrightarrow A \times B$ and $B \hookrightarrow A\times B$. Show that $K$ is free.
As usual, thanks in advance :)