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The principal congruence subgroup $\Gamma(n)$ of $SL(2,\mathbb Z)$ is a normal subgroup of $SL(2,\mathbb Z)$ with index $$[SL(2,\mathbb Z):\Gamma(n)]=n^3\prod_{p|n}(1-\tfrac{1}{p^2}),$$ where the product is over all primes dividing $n$. The quotient group is isomorphic to $SL(2,\mathbb Z)/\Gamma(n)\cong SL(2,\mathbb Z/n\mathbb Z)$. Is this group known to be isomorphic to a family of basic groups, such as $S_n$, $A_n$ or $D_n$?

For $n=2$ the quotient group must be non-abelian and of order $6$, and thus isomorphic to the symmetric group $S_3$. Is it known for $n\geq 3$?

El Rafu
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  • The alternating group $A_n$ is not even a quotient of $SL_2(\Bbb Z/N\Bbb Z)$ for any $N$ and any $n > 5$ (The group $SL_2(\Bbb Z/N\Bbb Z)$ is not isomorphic to such families of "basic groups". Look at the center). – Dietrich Burde Apr 20 '23 at 12:50
  • Thanks. It seems from this paper https://pjm.ppu.edu/sites/default/files/papers/PJM_April2018_521to526.pdf that the quotients are isomorphic to $S_3$, $A_4$, $S_4$, $A_5$ for $n=2,3,4,5$. But this doesn't continue for $n>5$. – El Rafu Apr 20 '23 at 12:57
  • In that paper they consider the quotients of the modular group $\Gamma$ (and not of $SL_2(\Bbb Z/N\Bbb Z)$), if I am not mistaken. – Dietrich Burde Apr 20 '23 at 13:01
  • Yes, I'm interested in the quotients $SL(2,\mathbb Z)/\Gamma(n)$. Is this not what you meant? – El Rafu Apr 20 '23 at 13:03
  • You wrote this: "$SL(2,\mathbb Z/n\mathbb Z)$. Is this group known to be isomorphic to a family of basic groups, such as $S_n$, $A_n$ or $D_n$?" So I assume you want to study the group $SL(2,\mathbb Z/N\mathbb Z)$ (where $N$ need not be $n$ in general). And this group is not isomorphic to $A_n,S_n,D_n$ in general. – Dietrich Burde Apr 20 '23 at 13:04
  • Yes indeed. But why am I then interested in quotients of $SL(2,\mathbb Z/n\mathbb Z)$? I was wondering if it is isomorphic to some finite group that I understand better, but even the index calculation would rule out $A_n$ or $S_n$ of course. – El Rafu Apr 20 '23 at 13:06
  • Let us continue this discussion in chat. – El Rafu Apr 20 '23 at 13:11

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