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"left invariant" -> "stable" (to avoid confusion with left vs. right), and then some more re-writing.
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edit approved Oct 1, 2016 at 21:51
LSpice
edit approved Oct 1, 2016 at 21:51
LSpice
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are the congruence subgroups $\Gamma(n)$ characteristic inside $SL_2$\mathrm{SL}_2(\mathbb{Z})$?

AreFor which $n$ is the "principal congruence subgroups"subgroup" $\Gamma(n)\le SL_2(\mathbb{Z})$$\Gamma(n)\le \mathrm{SL}_2(\mathbb{Z})$, the subgroup consisting of matrices congruent to the identity modulo $n$, characteristic?

Ie I.e., are they left invariantfor which $n$ is $\Gamma(n)$ stable (as a set) byunder all automorphisms of $SL_2(\mathbb{Z})$$\mathrm{SL}_2(\mathbb{Z})$?

Here $\Gamma(n)$ is the subgroup consisting of matrices congruent to the identity mod $n$.

are the congruence subgroups $\Gamma(n)$ characteristic inside $SL_2(\mathbb{Z})$?

Are the "principal congruence subgroups" $\Gamma(n)\le SL_2(\mathbb{Z})$ characteristic?

Ie, are they left invariant (as a set) by all automorphisms of $SL_2(\mathbb{Z})$?

Here $\Gamma(n)$ is the subgroup consisting of matrices congruent to the identity mod $n$.

are the congruence subgroups $\Gamma(n)$ characteristic inside $\mathrm{SL}_2(\mathbb{Z})$?

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 $\mathrm{SL}_2(\mathbb{Z})$?

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stupid_question_bot
stupid_question_bot
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are the congruence subgroups $\Gamma(n)$ characteristic inside $SL_2(\mathbb{Z})$?

Are the "principal congruence subgroups" $\Gamma(n)\le SL_2(\mathbb{Z})$ characteristic?

Ie, are they left invariant (as a set) by all automorphisms of $SL_2(\mathbb{Z})$?

Here $\Gamma(n)$ is the subgroup consisting of matrices congruent to the identity mod $n$.