For which $n$ is the "principal congruence subgroup" $\Gamma(n)\le \mathrm{SL}\_2(\mathbb{Z})$, the subgroup consisting of matrices congruent to the identity modulo $n$, characteristic? I.e., for which $n$ is $\Gamma(n)$ stable (as a set) under all automorphisms of $\mathrm{SL}\_2(\mathbb{Z})$?



are the congruence subgroups $\Gamma(n)$ characteristic inside $\mathrm{SL}_2(\mathbb{Z})$?
No for $n$ odd, yes for $n$ even.


By a result of [Hua and Reiner](http://www.ams.org/journals/tran/1951-071-03/S0002-9947-1951-0043847-X/home.html), the automorphism group of $SL\_2(\mathbb{Z})$ is generated by (1) conjugation by $GL\_2(\mathbb{Z})$ (the matrices with determinant $\pm 1$) and (2) the map $X \mapsto \epsilon(X) \cdot X$ where $\epsilon$ is the unique nontrivial character $SL\_2(\mathbb{Z}) \to \pm 1$. The maps of the form (1) clearly preserve $\Gamma(n)$, so we just need to check the single map $X \mapsto \epsilon(X) \cdot X$.


We can describe $\epsilon$ as follows: $SL\_2(\mathbb{Z})$ acts on the three points of $\mathbb{P}^1(\mathbb{F}\_2)$, giving a map $SL\_2(\mathbb{Z}) \to S\_3$. Compose this with the sign character $S\_3 \to \pm 1$.


If $n$ is odd, then $\left( \begin{smallmatrix} 1 & n \\ 0 & 1 \end{smallmatrix} \right) \equiv \left( \begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix} \right) \bmod 2$, so $\epsilon \left( \begin{smallmatrix} 1 & n \\ 0 & 1 \end{smallmatrix} \right) = -1$. So $\left( \begin{smallmatrix} 1 & n \\ 0 & 1 \end{smallmatrix} \right)$ is in $\Gamma(n)$ but $\epsilon \left( \begin{smallmatrix} 1 & n \\ 0 & 1 \end{smallmatrix} \right) \cdot \left( \begin{smallmatrix} 1 & n \\ 0 & 1 \end{smallmatrix} \right) = \left( \begin{smallmatrix} -1 & -n \\ 0 & -1 \end{smallmatrix} \right)$ is not.


On the other hand, if $n$ is even then $X \equiv \mathrm{Id} \bmod n$ shows that $X$ maps to $e$ in $S\_3$, so $\epsilon(X)=1$ and $\epsilon(X) X = X$ for all $X \in \Gamma(n)$.


