Let $\Gamma(N)$ be the principal congruence subgroup of level $N$ in $\mathrm{SL}\_n(\mathbf{Z})$, where $n\geq 3$. Then $\Gamma(N)$ is residually $p$-finite for all primes $p$ dividing $N$.



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> Can $\Gamma(N)$ be residually $p$-finite for any prime $p$ that does not divide $N$ ?
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On a related note: $\Gamma(N)$ is residually $p$-finite for only finitely many primes $p$. The proof I know is somewhat indirect: 1) (Rhemtulla) if a group is residually $p$-finite for infinitely many primes $p$, then it is orderable. 2) (Witte) no finite index subgroup of $\mathrm{SL}\_n(\mathbf{Z})$, where $n\geq 3$, is orderable. Is there a more direct / hands-on proof?



Residual $p$-finiteness of principal congruence subgroups
No. In fact, I claim that if $G$ is any solvable group and $\phi : \Gamma(N) \rightarrow G$ is a surjection, then $G$ is a finite group and all primes that divide $|G|$ also divide $N$. The key is the following beautiful theorem of Lee and Szczarba.



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> **Theorem:** If $n \geq 3$ and $\Gamma(N)$ is the level $N$ principal congruence subgroup of $\mathrm{SL}\_n(\mathbb{Z})$, then $[\Gamma(N),\Gamma(N)] = \Gamma(N^2)$.
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See their paper 


MR0422498 (54 #10485) 
Lee, Ronnie; Szczarba, R. H.
On the homology and cohomology of congruence subgroups. 
Invent. Math. 33 (1976), no. 1, 15–53. [DOI link](https://doi.org/10.1007/BF01425503)


Anyway, this implies that the derived series of $\Gamma(N)$ is
$$\Gamma(N) > \Gamma(N^2) > \Gamma(N^4) > \cdots.$$
Any surjection to a solvable group thus contains $\Gamma(N^{2^k})$ in its kernel for some $k$. But it also follows from Lee-Szczarba's work that $\Gamma(M)/\Gamma(M^2)$ is an abelian group all of whose elements have order $M$. This implies that all the primes which divide the order of $\Gamma(N) / \Gamma(N^{2^k})$ also divide $N$. The desired result follows.


