If $l$ is a prime number $p$, then I know how to compute the order of $SL_n(\mathbb Z_p)$. (By using row vectors linearly independent choices) But what would be the order if $l$ is any natural number(may not be a prime).
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For $n=2$ see here. Now generalize, using the Chinese Remainder Theorem. Actually, the answer is given there already. One can also start here. – Dietrich Burde Jan 30 '23 at 15:11
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See: https://en.wikipedia.org/wiki/Jordan%27s_totient_function#Order_of_matrix_groups – Samuel Adrian Antz Jan 30 '23 at 15:18
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$|GL_n(\Bbb{Z}/(l))|=\prod_{p^k|l} |GL_n(\Bbb{Z}/(p^k))|=\prod_{p^k|l} p^{n(k-1)} |GL_n(\Bbb{Z}/(p))|$ and $|GL_n(\Bbb{Z}/(l))|=\phi(l) |SL_n(\Bbb{Z}/(l))|$ – reuns Jan 30 '23 at 16:40