# Are these two quotient groups of $\mathbb{Z}^2$ isomorphic to each other?

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.
(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.) –  user29743 Jun 21 '13 at 3:05
(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.) –  anon Jun 21 '13 at 3:07
@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 $\mathbb{Z}/a\mathbb{Z} \oplus \mathbb{Z}/d\mathbb{Z}$ –  user29743 Jun 21 '13 at 3:09