# Re: [GAP Forum] An issue with finitely presented groups

```Dear Forum, Dear Dr. Rajeevsarathy,

The subgroup you give cannot have index 2, as it is easy to find larger index
in finite quotient images. (In fact I think the index is infinite.) Is there
possibly a misprint in group or realtors, or why do you think the index should
be 2?```
```
(The error message you get is indicating that coset enumeration did not
succeed. One can type `return;’ to try with more memory, but of course it can
never terminate  if the index is infinite.)

Regards,

Alexander Hulpke

Mitto ab tabulariUM meum

> On Apr 3, 2018, at 16:19, Dr. Kashyap Rajeevsarathy <kash...@iiserb.ac.in>
> wrote:
>
> Dear Forum,
>
> I was trying to ascertain the membership of a word (in this case a letter
> b2) in the index 2 subgroup of a finitely presented (surface) group. Please
> point me to a better alternative (if possible) to the example below:
>
>
> gap> f := FreeGroup("a1","b1","a2","b2");
>
> <free group on the generators [ a1, b1, a2, b2 ]>
>
> gap> genf := GeneratorsOfGroup(f);
>
> [ a1, b1, a2, b2 ]
>
> gap> G :=
> f/[genf[1]*genf[2]*genf[1]^(-1)*genf[2]^(-1)*genf[3]*genf[4]*genf[3]^(-1)*genf[4]^(-1)];
>
>
> <fp group of size infinity on the generators [ a1, b1, a2, b2 ]>
>
> gap> genG := GeneratorsOfGroup(G);
>
> [ a1, b1, a2, b2 ]
>
> gap> H := Subgroup(G,[genG[1],genG[2],genG[3],genG[4]^2]);
>
> Group([ a1, b1, a2, b2^2 ])
>
> gap> genH := GeneratorsOfGroup(H);
>
> [ a1, b1, a2, b2^2 ]
>
> gap> genG[4] in H;
>
> Error, the coset enumeration has defined more than 4096000 cosets
>
> called from
>
> TCENUM.CosetTableFromGensAndRels( fgens, grels, fsgens ) called from
>
> CosetTableFromGensAndRels( fgens, grels, fsgens ) called from
>
> TryCosetTableInWholeGroup( H ) called from
>
> CosetTableInWholeGroup( U ) called from
>
> <function "unknown">( <arguments> )
>
> called from read-eval loop at line 8 of *stdin*
>
> type 'return;' if you want to continue with a new limit of 8192000 cosets,
>
> type 'quit;' if you want to quit the coset enumeration,
>
> type 'maxlimit := 0; return;' in order to continue without a limit
>
> brk>
>
> Thanks,
> Kashyap
>
> --
> Kashyap Rajeevsarathy
> Assistant Professor,
> Indian Institute of Science Education and Research (IISER) Bhopal,
> Bhauri, Bhopal - 462066,
> Phone: +91-755-669-1273
> Website: https://home.iiserb.ac.in/~kashyap
> _______________________________________________
> Forum mailing list
> Forum@gap-system.org
> https://mail.gap-system.org/mailman/listinfo/forum

_______________________________________________
Forum mailing list
Forum@gap-system.org
https://mail.gap-system.org/mailman/listinfo/forum
```