On $SL(2, \mathbb{Z})$ and $SL(2, \mathbb{Z} / N \mathbb{Z})$ - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-18T06:02:15Z http://mathoverflow.net/feeds/question/57443 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/57443/on-sl2-mathbbz-and-sl2-mathbbz-n-mathbbz On $SL(2, \mathbb{Z})$ and $SL(2, \mathbb{Z} / N \mathbb{Z})$ eltonjohn 2011-03-05T09:50:29Z 2011-03-05T09:50:29Z <p>I have been told that there is an epimorphism $SL(2, \mathbb{Z}) \to SL(2, \mathbb{Z} / N \mathbb{Z})$, $N \in \mathbb{Z}$.</p> <p>I guess the epimophism can be constructed by</p> <p>1) Regard $x \in SL(2, \mathbb{Z} / N \mathbb{Z})$ as $x \in M(2, \mathbb{Z}) \cap GL(2, \mathbb{R})$,</p> <p>2) Apply a series of fundamental transformations on $x$ by elements of $SL(2, \mathbb{Z})$ to get an element in $SL(2, \mathbb{Z})$.</p> <p>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.</p> <p>Thanks in advance for any suggestion, as I have no privilege to add and/or modify comments here. </p>