Ie, if $\Gamma\subseteq\text{SL}_2(\mathbb{Z})$ is a normal subgroup, and $\alpha\in\text{GL}_2(\mathbb{Q})\cap M_2(\mathbb{Z})$, then is $\alpha\Gamma = \Gamma\alpha$?
(if necessary we can assume $\alpha$ has positive determinant).
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Ah, you just changed your question, so I have to update my answer. – 23rd Apr 24 '13 at 13:52
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yeah, the title question was correct, but I made a mistake in the actual post. – oxeimon Apr 24 '13 at 13:54
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Sorry, I didn't read your title carefully. – 23rd Apr 24 '13 at 14:24
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The answer is NO.
For example, let $\Gamma=SL(2,\mathbb{Z})$ and let $\alpha=\begin{pmatrix}2&0\\0&1\end{pmatrix}\in GL(2,\mathbb{Q})\cap M_2(\mathbb{Z})$. Then for any $\beta\in\alpha\Gamma$, $\beta_{12}$ must be even, but this is not true for $\Gamma\alpha$.
23rd
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