I have been told that there is an epimorphism $SL(2, \mathbb{Z}) \to SL(2, \mathbb{Z} / N \mathbb{Z})$, $N \in \mathbb{Z}$.
I guess the epimophism can be constructed by
1) Regard $x \in SL(2, \mathbb{Z} / N \mathbb{Z})$ as $x \in M(2, \mathbb{Z}) \cap GL(2, \mathbb{R})$,
2) Apply a series of fundamental transformations on $x$ by elements of $SL(2, \mathbb{Z})$ to get an element in $SL(2, \mathbb{Z})$.
My question is, is this a routine that can be dreamed up by anyone? All the documents I know say this is 'easy', giving no clues as to how to perform it.
Thanks in advance for any suggestion, as I have no privilege to add and/or modify comments here.
