[Principal congruence subgroups of $SL(n, \mathbb{Z})$](https://mathoverflow.net/questions/62446/principal-congruence-subgroups-of-sln-mathbbz)
 gives a discription of automorphisms of $SL(n,\mathbb Z)$. Is it true for n even too?. Hau-Reiner's paper gives generators for the group of automorphisms of $SL(n,\mathbb Z)$ which are induced by the automorphisms of $GL(n,\mathbb Z)$. 



Principal congruence subgroups of SL(n,Z)
Yes, I think the statement is also true for $n >2$ even (see Guntram's answer of the question [Automorphisms of $SL\_n(\mathbb{Z})$](https://mathoverflow.net/questions/57235/automorphisms-of-sl-n-mathbbz)). 


As a second reference a paper of O'Meara may serve. It can be found [here](http://www.digizeitschriften.de/dms/img/?PPN=PPN243919689_0223&DMDID=dmdlog7). The following specializes O'Meara's result to the case $SL\_n(\mathbb{Z})$: 


1. Note that in our context $\mathbb{o} = \mathbb{Z}, K = \mathbb{Q}, M = \mathbb{Z}^n.$.
2. Since the only automorphisms of the field $\mathbb{Q}$ is the identity, semi-linear here means linear.
3. By 1.13, $RL\_n(M)$ is the center of $GL\_n(\mathbb{Z})$, i.e. $RL\_n(M) = \lbrace I\_n, -I\_n \rbrace \cong \mathbb{Z}/2\mathbb{Z}$ and since $SL\_n(\mathbb{Z})$ is a perfect group, the only homomorphism $SL\_n(\mathbb{Z}) \to RL\_n(M)$ is the trivial one. Hence the radical automorphisms $P\_{\chi}$ in 5.4 Theorem A are just the identity.
4. By 5.4 Theorem A it follows that the automorphism of $SL\_n(\mathbb{Z})$ are of the form $\Phi\_g, \Psi\_h$ that are given in matrix form (end of page 87) by $$S \mapsto PSP^{-1}, S \mapsto P^{-1}(S^T)^{-1}P$$ for some $P \in GL\_n(\mathbb{Q})$.
5. By 5.5 Theorem B, $\Phi\_g$ is an automorphism of $SL\_n(\mathbb{Z})$ iff $P\*\mathbb{Z}^n = q\mathbb{Z}^n$ for some $q \in \mathbb{Q}$ (where $\*$ is matrix-vector multiplication). It's not hard to see that this implies $P = qQ$ for some $Q \in GL\_n(\mathbb{Z})$. Thus $\Phi\_g: S \mapsto QSQ^{-1}$.
6. By 5.6 Theorem C, $\Psi\_h$ is an automorphism of $SL\_n(\mathbb{Z})$ iff $h(M) = qM^\ast$ for some $q \in \mathbb{Q}$, where $M^\*$ is the free $\mathbb{Z}$-module whose base is the dual base of the standard vectors in $\mathbb{Q}^n$ (see page 65). If $P$ is the matrix of $h$ in the standard base of $\mathbb{Q}^n$ and its dual base in $(\mathbb{Q}^n)^\ast$ then, again, it's not hard to see that $P = qQ$ for some $Q \in GL\_n(\mathbb{Z})$. Thus $\Psi\_h: S \mapsto Q^{-1}(S^T)^{-1}Q$.


In particular each automorphism of $SL\_n(\mathbb{Z})$ is induced by an automorphism of $GL\_n(\mathbb{Z})$ (what was left open in the Hua-Reiner paper). 


