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So i have group $G=(M_2(\mathbb{Z}),+)$ and $H=\{h\in G \mid \mathrm{tr}(h)=0\}$.

So i already proved : H $\triangleleft$ G

Now i need some help when i'm trying to find what kind of elements are there in $G/H$ as in quotient group. And how can i find an isomorphism between $G/H$ and $\mathbb{Z}$.

So any help would be useful and deeply appreciated.

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We try to use the first isomorphism theorem. We have a homomorphism $\theta: G\to \mathbb{Z}$ defined by $A\mapsto\mathrm{tr}(A)$ and then $H=\ker(\theta)$. Trivially, our homomorphism is a surjective homomorphism (just take a matrix with the top left being $a\in \mathbb{Z}$ with $0$ in the other entries to show surjectivity) so $G/H\cong \mathbb{Z}$.

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