Is there an easy way to see that $\mathbf{GL}_2(\mathbb{Z})$ and $\mathbf{GL}_2(\mathbb{Q})$ are not isomorphic ?
2026-04-02 04:36:41.1775104601
Morphisms between $\mathbf{GL}_2(\mathbb{Z})$ and $\mathbf{GL}_2(\mathbb{Q})$
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Notice that the center of $\mathrm{GL}_2(\mathbf{Z})$ is $\{I, -I\}$, whereas the center of $\mathrm{GL}_2(\mathbf{Q})$ is $\{qI : q \in \mathbf{Q}^\times\}$, which is infinite.