# Proof that principal congruence subgroups $\Gamma(N)$ are torsion free.

I have read the fact that the principal congruence subgroups $$\Gamma(N)$$ of $$\mathrm{GL}_n({\mathbb{Z}})$$ are torsion free for $$N \geq 3$$ several times, but only saw proofs for very specific situations. Could someone give me a hint where to find a general proof?

I am interested in this, because i need something similar in a situation where this fact can not be applied directly and I want to see if the proof still works. That is, for an imaginary quadratic number field $$\mathbb{Q}(\sqrt{-k})$$ and the corresponding algebraic group $$\mathrm{SU}(n,1)$$, one can attach an arithmetic subgroup $$\Gamma$$ to a choice of parahoric Subgroups $$P_v$$ for all nonarchemedian places $$v$$. However, this $$\Gamma$$ can not be imbedded into some $$\mathrm{GL}_m(\mathbb{Z})$$, but rather into $$\mathrm{GL}_m(\mathbb{Z}[\frac{1}{k}])$$. Is there a reasonable way to define $$\Gamma(N)$$, for example, when $$N$$ and $$k$$ are coprime, as the kernel of the reduction modulo $$N$$ map (where $$\frac{1}{k}$$ is just considere as $$k^{-1}$$ modulo $$N$$), such that the groups $$\Gamma(N)$$ are torsion free (for $$N\geq 3$$) and normal in $$\Gamma$$? Help would be much appreciated!

First observe that $$\Gamma(p^l) \subseteq \Gamma(p)$$ for any prime number $$p$$ and any $$l \geq 0$$. For prime numbers $$p \geq 3$$ the group $$\Gamma(p)$$ is torsion-free. See this: https://math.stackexchange.com/q/3979175.
Now, $$N = \prod_{p| N} p^{\nu_p}$$ ($$p$$ prime number) and by Chinese remainder theorem it is $$\mathbb{Z}/N\mathbb{Z} \cong \prod_{p | N} \mathbb{Z}/p^{\nu_p}\mathbb{Z}$$ and therefore $$\Gamma(N) = \bigcap_{p | N}\Gamma(p^{\nu_p})$$. If there is some prime $$p \geq 3$$ with $$p | N$$, then $$\Gamma(N) \subseteq \Gamma(p^{\nu_p}) \subseteq \Gamma(p)$$. Since $$\Gamma(p)$$ is torsion-free, $$\Gamma(N) \subseteq \Gamma(p)$$ is torsion-free.
So, the only thing remaining to prove is that $$\Gamma(4)$$ is torsion-free. This is also at the end of https://math.stackexchange.com/q/3979175.