Welcome to MathOverflow

Question

7

3

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

flag	asked Apr 20 2011 at 18:33 Poove 195●5

Ralph · Accepted Answer · 2011-04-20 20:16:07Z UTC

8

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 http://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).

answered Apr 20 2011 at 20:16

Ralph
12.5k●10●36

1

This is exactly the same answer as @Richard's – Igor Rivin Apr 20 2011 at 21:57

3

Yes it is. When I was starting to type there was no answer and after I saved mine, I saw Richard's. So I guess there was some overlapping in time. Nevertheless, I decided no to delete my answer since a) it confirms what Richard says he thinks and b) it has some more details about the generators (and the case $n = 2$) that might be interesting for readers who just overflow the question without looking into the mathoverflow link or the Hua-Reiner paper. – Ralph Apr 20 2011 at 22:13

you are right Ralph – Poove Apr 21 2011 at 5:56

@Ralph: agreed. – Igor Rivin Apr 25 2011 at 1:25

@Ralph: $\Gamma_n$ is characteristic. but what you said about the automorphism group of $SL(n,\mathbb Z)$ is true for n odd, because for n odd, all automorphisms of $SL(n,\mathbb Z)$ are induced by automorphisms of $GL(n,\mathbb Z)$(Hua-Reiner,theorem.3). Is it true if n even too? – Poove Apr 27 2011 at 9:30

show 3 more comments

Richard Kent · Answer 2 · 2011-04-20 19:56:53Z UTC

10

I think the answer is yes.

See this question for a discussion of $\mathrm{Aut}(\mathrm{SL}(n,\mathbb{Z}))$. The generators given in the Hua-Reiner paper seem to preserve the principal congruence subgroups.

answered Apr 20 2011 at 19:56

Richard Kent
8,525●2●29●58

Thanks a lot Richard. – Poove Apr 21 2011 at 5:55

Principal congruence subgroups of $SL(n, \mathbb{Z})$

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

2 Answers

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.