> < ^ Date: Tue, 03 Sep 1996 16:45:49 -0300
< ^ From: Norai R. Rocco <norai@mat.unb.br >
> ^ Subject: small problem in coset enumeration

Dear GAP Forum,

I haven't been able to understand the following situation in the coset
enumeration:

Given, for instance, the finitely presented (finite) group g,

```gap> F:=FreeGroup("a", "b");;
gap>
gap> g:=F/[F.1^4, F.2^4, Comm(F.1, F.2)^F.1/Comm(F.1, F.2^(-1)),
Comm(F.1, F.2)^F.2/Comm(F.1^(-1), F.2),
Comm(F.1^2, F.2)/Comm(F.1, F.2^(-1))*Comm(F.1, F.2)];
Group( a, b )
gap>
```

then GAP computes its Size quickly:

gap> Index(g, TrivialSubgroup(g));
128

Now observe that the last relator comes from the commutator identity

```[x^2, y] = [x, y]^x*[x, y]
```

which should give the last relator as a consequence of the third one

```[F.1^2, F.2] = [F.1, F.2^(-1)]*[F.1, F.2]
```

However, removing the last relator one gets the presentation

```gap> h:=F/[F.1^4, F.2^4, Comm(F.1, F.2)^F.1/Comm(F.1, F.2^(-1)),
Comm(F.1, F.2)^F.2/Comm(F.1^(-1), F.2)];
Group( a, b )
gap>
```

and now,

gap> Index(h, TrivialSubgroup(h));
Error, the coset enumeration has defined more than 64000 cosets:
type 'return;' if you want to continue with a new limit of 128000 cosets,
type 'quit;' if you want to quit the coset enumeration,
type 'maxlimit := 0; return;' in order to continue without a limit,
in
CosetTableFpGroup( G, H ) called from
G.operations.Index( G, U ) called from
Index( h, TrivialSubgroup( h ) ) called from
main loop
brk> quit;
gap> quit;

Could anyone tell me what's wrong? Seems to be a small bug in coset
enumaration (?).

Thanks in advance for any help.
Norai Rocco.

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