> < ^ Date: Wed, 04 Sep 1996 17:37:53 +1000
^ From: George Havas <havas@cs.uq.edu.au >
< ^ Subject: Re: small problem in coset enumeration

On Tue, 3 Sep 1996, Norai Romeu Rocco wrote:

```>   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]
>
...............
>   Could anyone tell me what's wrong? Seems to be a small bug in coset
>   enumaration (?).
>
NO --- User error: missing parentheses in
```
```Comm(F.1^2, F.2)/Comm(F.1, F.2^(-1))*Comm(F.1, F.2)];
```

should be:

```Comm(F.1^2, F.2)/ ( Comm(F.1, F.2^(-1))*Comm(F.1, F.2)] );
```

/ and = are not equivalent operators in presentations.

I think this explains it all.

```George Havas (havas@cs.uq.edu.au) Phone: +61 7 3365 2904; Fax: +61 7 3365 1999
Department of Computer Science    AUS    http://www.cs.uq.oz.au/personal/havas
The University of Queensland      UK http://www-groups.dcs.st-and.ac.uk/~havas
Queensland 4072 AUSTRALIA         US          http://dimacs.rutgers.edu/~havas
```

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