I have troubles to solve this kind of exercises. For example:
Let $$G_1=\langle x,y |x^3=y^4=1\rangle,~~~G_2=\langle x,y |x^6=y^6=(xy)^3=1\rangle. $$ I want to check that $G_1$ is an infinite nonabelian group and in $G_2$ we have $xy^2x \neq 1$ .
For the first part, I have seen that it is useful to define a group homomorphism and then see that the image (that we know) is infinite and nonabelian. For the second part, similarly we can define a $\phi$ such that $\phi (xy^2x) \neq 1$.
How can I define this homomorphisms? There is any general procedure for this?