Let $n, m > 1$. The map $\mathbb{Z} \twoheadrightarrow \mathbb{Z}/m\mathbb{Z}$, of reduction mod $m$, induces a group homomorphism $F: \text{SL}_n(\mathbb{Z}) \to \text{SL}_n(\mathbb{Z}/m\mathbb{Z})$. My question is, is $F$ surjective or not?
2026-04-24 02:24:17.1776997457
Induced group homomorphism $\text{SL}_n(\mathbb{Z}) \twoheadrightarrow \text{SL}_n(\mathbb{Z}/m\mathbb{Z})$ surjective?
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This turned out to be a non-example. I'm leaving it up for posterity as a CW answer.
As it turns out, the answer is no. The smallest counterexample I can think of is as follows: we note that $$ \pmatrix{2&1&1\\1&2&1\\1&1&2} \in SL_n(\Bbb Z/3\Bbb Z) $$ However, this matrix is not the reduction mod $3$ of any element of $SL_n(\Bbb Z)$.