TITLE: Presentation of Groups
QUESTION [7 upvotes]: 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? 
Thanks,

REPLY [2 votes]: Let $H = {\rm SL}(2,{\mathbb Z})$ and define $\phi:G_1 \to H$ by
$$ \phi(x) = \left(\begin{array}{rr}0&1\\-1&-1\end{array}\right),
\phi(y) = \left(\begin{array}{rr}0&1\\-1&0\end{array}\right).$$
Let $K=S_6$ and define $\psi:G_2 \to K$ by
$$\psi(x)=(1,2,3,4,5,6),\,\psi(y)=(1, 2, 6, 5, 4, 3).$$