# Database subgroups of free group

Is there some database that contains "all" low-index normal subgroups of the free group on two generators?

Extension: does there exist such a GAP-database?

Thank you!

• Described as what? (I mean, they are all abstractly isomorphic, but on the other hand, the quotients they give include all finite simple groups, depending on what you mean by "low"). Sep 25 '18 at 10:24
• I would like them described by their generators. So I can impose extra demands on them and prune to find normal subgroups of a certain form... Sep 25 '18 at 11:21

I think the answer is no. There exists a Magma command $$\mathtt{LowIndexNormalSubgroups}$$ that does what you want, and it does indeed find generators for each of the subgroups. I believe that a version in $$\mathsf{GAP}$$ is being written and will be available shortly.

Using Magma I got up to index 50 without having to wait too long, and you could get up to indexes at least 70 or 80 by waiting longer. If you want to do a lot of computations with these normal subgroups then it would be worthwhile doing a single long run and storing the result.

Here is my Magma run.

> F := FreeGroup(2);
> time #LowIndexNormalSubgroups(F,10);
97
Time: 1.750
> time #LowIndexNormalSubgroups(F,20);
425
Time: 8.760
> time #LowIndexNormalSubgroups(F,30);
999
Time: 30.510
> time #LowIndexNormalSubgroups(F,40);
1844
Time: 118.260
> time #LowIndexNormalSubgroups(F,50);
3058
Time: 573.920