I want to know whether the principal congruence subgroups of $SL(n, \mathbb{Z})$ are characteristic? please suggest me a reference.



Principal congruence subgroups of $SL(n, \mathbb{Z})$
Yes, they are caracteristic. 


Let $\rho: GL(n, \mathbb{Z}) \to GL(n, \mathbb{Z}/m\mathbb{Z})$ be reduction mod $m$ and $\Gamma\_n(m) := ker(\rho) \cap SL(n, \mathbb{Z})$ be a congruence sugroup. 


According to the discussion in [Automorphisms of $SL\_n(\mathbb{Z})$](https://mathoverflow.net/questions/57235/automorphisms-of-sl-n-mathbbz) the automorphisms of $SL(n, \mathbb{Z})$ for $n > 2$ are generated by 


* $X \mapsto (X^T)^{-1}$
* $X \mapsto AXA^{-1}, A \in GL(n, \mathbb{Z}).$


Obviously, $\Gamma\_n(m)$ is invariant under these automorphisms and thus is characteristic. 


In case $n = 2$ there is one more automorphisms of $SL(2, \mathbb{Z})$ that also leaves $\Gamma\_n(m)$ invariant (for a description of this automorphism see the paper of Hua and Reiner mentioned in the link above). 


