Let $N\geq2$ be a positive integer. Is the canonical homomorphism $\pi$ from $SL\_2(\mathbb{Z})$ to $SL\_2(\mathbb{Z}/N\mathbb{Z})$ surjective?


What if we ask the same question for $SL\_n$?



Modular group modulo $N$
Yes, it is surjective. It is called the Strong Approximation Property (because of a more general way to formulate it). In your case, there is a simple proof: $SL\_n(\mathbb{Z})$ contains the elementary matrices, and their projections generate $SL\_n(\mathbb{Z}/N\mathbb{Z})$ because $\mathbb{Z}/N\mathbb{Z}$ is Euclidean.


