Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Suppose I have a subgroup of $\textrm{SL}_2(\mathbb Z)$ given by 3 generators, and it happens to be of finite index in $\textrm{SL}_2$. Is there a way (on Sage, since that is what I have access to) to figure out its index? I tried defining a matrix group via the generators and using index(), but this gave me an error. Perhaps if I define the group via permutations it would work? I am new to Sage, so any help would be appreciated.

share|improve this question
    
Have you tried asking your question in the askSAGE forum? –  J.C. Ottem Apr 28 '11 at 17:14
    
No, I wasn't aware of it -- I'll try it there. –  Elena Apr 28 '11 at 17:18

2 Answers 2

To add to my comments on @Emmanuel's answer: first, compute the intersection with $\Gamma(2).$ This is quite easy, since $SL(2, 2)$ is not a very large group, and computing matrices mod 2 is not difficult either. Express the generators in words in a generating set of $\Gamma(2)$ (the usual parabolic generators corresponding to $z\rightarrow z + 2$ and $z\rightarrow \dfrac{z}{z+2}$ are always good). Then use

www.gap-system.org/Manuals/pkg/fga/doc/manual.pdf (if @Emmanuel is correct, you can use this from sage. If not, gap is free also...)

Alternatively, you can post your three generators...

share|improve this answer
    
I thought you meant that one could use the Schreier index formula for subgroups of free groups. I could post the generators but I think it will be useful for me to work this out for myself! –  Elena Apr 29 '11 at 15:43
    
I did mean that also, but notice that there are plenty of two-generator subgroups of $F_2$ of infinite index, so that works only when the group is of finite index... –  Igor Rivin Apr 29 '11 at 16:10
    
Exactly! Which is why I made the comment that I thought to do what you said requires you to know that your group is finite index :) But the gap thing is something I should try. –  Elena Apr 29 '11 at 16:14

I don't know about Sage (which presumably would use Gap for such a thing?) but according to its documentation (I haven't tried), Magma has an algorithm due to Coxeter-Todd to (attempt to) find coset representatives for a subgroup of a finitely presented group. Assuming you can represent your generators in terms of some standard generators of SL_2(Z), that might give you the index. See here:

http://magma.maths.usyd.edu.au/magma/handbook/text/739#8205

share|improve this answer
    
Todd-Coxeter attempts to solve an undecidable problem, so is not going to be fast. $\Gamma(2)$ is a free subgroup of finite index; computing generators of intersection of your subgroup with $\Gamma_2$ is easy, then you just count how many there are, and that will give you the index. –  Igor Rivin Apr 28 '11 at 17:32
    
Igor -- but then I guess I would really have to be sure the group is finite index (in this case I just think it should be but haven't really confirmed it). That's why I was hoping to use some computer program to confirm it first. By the way, do you have any experience with the Todd-Coxeter algorithm wrt its slowness? Emmanuel -- I am aware of the Magma algorithm, but I don't have access to Magma at the moment. Thanks for the suggestion, though. –  Elena Apr 28 '11 at 18:14
    
@Elena: you don't need to be sure the subgroup is of finite index. Whether or not it is, you will compute a finite index free subgroup of it, which will be a finitely generated subgroup of a free group. Computing its index is easy (whether or not its finite). –  Igor Rivin Apr 28 '11 at 18:22

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.