proving group is infinite

Question

Consider the group $G=\langle a,b \mid a^2=b^3=1 \rangle$.
I would like to show that $G$ is infinite by finding a homomorphism $\phi :G \rightarrow \mathbb{Z}$ and showing that this homomorphism is surjective. I am not sure how to define the map $\phi$.

Answer 1

Suppose there was a homomorphism $f\colon G\to\mathbb{Z}$ like that. Since the only integer with a finite order is 0, it follows that $f(a)=0=f(b)$ because $f(a)^2=0=f(b)^3$. Then your surjective homomorphism would be constantly 0, and this is a contradiction.