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I am trying to tell if two quotient groups of $\mathbb{Z}^2$ are isomorphic.

Let $H$ be the subgroup generated by $\{(1, 3),(1, 7)\}$ and $G$ the subgroup generated by $\{(2, 4),(2, 6)\}$. Are the quotient groups $\mathbb{Z}^2/H$ and $\mathbb{Z}^2/G$ isomorphic?

I feel like this will be no with having something to do with relatively prime but I am not sure how to prove this. Thanks!

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Hint: invertible $\mathbb{Z}$-linear row operations to the matrix $$ \pmatrix{ a & b \\ c & d } $$ do not change the row space of the matrix. Use the Euclidean algorithm to row reduce your matrices to diagonal matrices, then conclude.

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  • $\begingroup$ (in case you've only done linear algebra over fields and not "linear algebra" over $\mathbb{Z}$ before, note that you can't multiply a row by a scalar other than $\pm 1$, but you can still add a scalar multiple of one row to another row.) $\endgroup$ – user29743 Jun 21 '13 at 3:05
  • $\begingroup$ (In case it isn't clear to readers, the row space of an integer matrix is the same as the subgroup of ${\bf Z}^2$ generated by the row vectors.) $\endgroup$ – anon Jun 21 '13 at 3:07
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    $\begingroup$ @anon thanks! i also should say the reason we want the matrices to be diagonal is because $\mathbb{Z}^2/\langle (a, 0), (0, d) \rangle$ is isomorphic to $\mathbb{Z}/a\mathbb{Z} \oplus \mathbb{Z}/d\mathbb{Z}$ $\endgroup$ – user29743 Jun 21 '13 at 3:09

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