2

EDIT: Let $N\geq 2$ be a natural number. Consider the natural group homomorphism $$SL_2(\mathbb{Z})\to SL_2(\mathbb{Z}/N\mathbb{Z})$$ given by reduction modulo $N$.

Is it onto? If not is anything known about its image?

MKO
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    Hint: what happens to the determinant? – Aphelli Mar 28 '23 at 10:54
  • Oops... Got it. Thanks! Then probably I have to modify the question and replace $GL_2$ by $SL_2$. – MKO Mar 28 '23 at 11:20
  • Then yes, this map is well-known to be onto. This was probably asked on this website, too. – Aphelli Mar 28 '23 at 12:51
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    Indeed, one of the related questions shown is https://math.stackexchange.com/questions/1409197/induced-group-homomorphism-textsl-n-mathbbz-twoheadrightarrow-textsl – mr_e_man Mar 28 '23 at 13:51
  • Given $\pmatrix{a&0\0&d} \in SL_2(\Bbb{Z}/N\Bbb{Z})$ It is not hard to find a matrix in $SL_2(\Bbb{Z})$ it is the reduction of. Then check that $\pmatrix{a&0\0&d},\pmatrix{1&1\0&1},\pmatrix{1&0\1&1}$ generate $SL_2(\Bbb{Z}/N\Bbb{Z})$. – reuns Mar 28 '23 at 17:16

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