(a) Given the general linear group (over the integers) GL(3,Z), show the subgroup H generated by $\begin{bmatrix}0 & 1 & 0\\ -1 & -1 & 0\\ 0 & 0 & 1\end{bmatrix}$ and $\begin{bmatrix}0 & 1 & 1\\ -1 & -1 & 0\\ 0 & 0 & 1\end{bmatrix}$ is infinite.
(b) Given the presented group G_3=
My approach: For the firsth part of the problem I aim to construct a sequence of infinite matrices in H, but all sequences I tried terminated to be the identity matrix.
For the second part it's clear that the given generators of H satisfy the relations of G, so by Von Dyck's theorem H is aquotient of $ G_3$, but does this guarentee the existence of such surjective homomorpshism? In general, if given an abstractly presented group with relations which are all satisfied by generators of another known group, is there always a surjective homomorphism between the two groups which preserves the corresponding generators?