Homomorphisms between $\mathbb{Z} \star \mathbb{Z} $ and $\mathbb{Z} _2 $

Question

What does a homomorphism between the free product $\mathbb{Z} \star \mathbb{Z} $ and $\mathbb{Z}_2 $ look like? I'm having trouble trying to do anything with this.

Answer 1

Think about the epimorphism $\Bbb Z\rightarrow \Bbb Z_2: x\mapsto x\mod 2$.

For $\Bbb Z^2\rightarrow\Bbb Z_2$ there are two obvious possibilities

$(x,y)\mapsto x+y \mod 2$ and $(x,y)\mapsto x\cdot y \mod 2$

since $\mod n$, $n\geq 2$, is a congruence relation, i.e. compatible with the operations of addition and multiplication.