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!



Database subgroups of free group
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

```

