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I wish to calculate the $$\langle x,y,z\mid 2x=2y=2z\rangle$$ generated over $\mathbb{Z}$, as a finitely generated $\mathbb{Z}$-module with the relations $2x=2y=2z$.

I tried to rewrite it as $$\langle x-y,y-z,z\mid 2(x-y)=0, 2(y-z)=0\rangle\cong\mathbb{Z}\oplus\mathbb{Z}/2\oplus\mathbb{Z}/2.$$

Is the working correct? Thanks for verification/correction!

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This is indeed the right approach. What you are really doing is computing the Smith normal form of the coefficient matrix of the relations. For example in your case the smith normal form of $\begin{pmatrix}2& -2&0\\0&2&-2\end{pmatrix}$ is $\begin{pmatrix}2& 0&0\\0&2&0\end{pmatrix}$, which implies that the group you have is indeed $\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}$. For computing abelian groups given by generators and relations, performing base change results in a similar relations matrix, and by the Smith normal form we can choose the base change such that the resulting matrix is diagonal, from which the group can be computed easily.

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