## On $SL(2, \mathbb{Z})$ and $SL(2, \mathbb{Z} / N \mathbb{Z})$ [closed]

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.

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 This is a standard homework exercise, so I'm voting to close; for a sketch of the argument, see this problem set: math.upenn.edu/~ted/620F09/homework/620hw5.pdf In any case, the general keyword to search on for this sort of question is "strong approximation." – Daniel Litt Mar 5 2011 at 10:11 Thank you very much for the information. Yes, I will search for "strong approximation". -eltonjohn – eltonjohn Mar 5 2011 at 10:43