Not everything is (only) abstract nonsense

Prove that \mathrm{SL}_2(\mathbb{Z})\to\mathrm{SL}_2(\mathbb{Z}/n\mathbb{Z}) is surjective for every integer n but \mathrm{GL}_2(\mathbb{Z})\to\mathrm{GL}_2(\mathbb{Z}/n\mathbb{Z}) is not necessarily surjective.

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Not everything is (only) abstract nonsense

1 is to q as n! is to…

If one believes that [n]=\{1,\dots,n\} is a vector space over the field with one element \mathbb{F}_1, then a permutation of [n] is just a complete flag. Concretely, a permutation (x_{\sigma(1)},\dots,x_{\sigma(n)}) corresponds to the chain of subspaces \{x_{\sigma(1)}\}\subseteq \{x_{\sigma(1)}, x_{\sigma(2)}\}\subseteq\dots\subseteq \{x_{\sigma_1}, x_{\sigma(2)},\dots,x_{\sigma(n)}\}.

As we all know, there are n! permutations of [n], or equivalently complete flags of \mathbb{F}_1^n. Give a formula for the number of complete flags of \mathbb{F}_q^n, which will be a q-analogue for the factorial.

1 is to q as n! is to…