> < ^ Date: Thu, 05 Jun 1997 09:37:43 +1000
> < ^ From: Burkhard Hoefling <b.hoefling@tu-bs.de >
> < ^ Subject: Re: Generating a Character Table

Dear Forum,

in his recent message to the forum, Mark Lewis described a problem computing=
with Ag groups obtained via AgGroupFpGroup from a finitely presented group.=
=20
His problem is due to the fact that the particular Ag representation is not=
*consistent*; as the section about "AgGroupFpGroup" in the GAP manual=
points out, this may lead to wrong computations; as it seems, it may also=

Try the following:

```gap> c := AbstractGenerator ("c");;
gap> d := AbstractGenerator ("d");;
gap> a1 := AbstractGenerator ("a1");;
gap> a2 := AbstractGenerator ("a2");;
gap> b1 := AbstractGenerator ("b1");;
gap> b2 := AbstractGenerator ("b2");;
gap>=20
gap> #construct Mark's group
gap> g3 := AgGroupFpGroup( rec(  =20
>     generators := [d,c,a1,a2,b1,b2],
>     relators := [b1^2,b2^2,a1^2*(b1)^(-1),a2^2*(b2)^(-1),c^3,d^2,
>                 (a1^c)*(a2)^(-1),(a2^c)*(a1*a2)^(-1),
>                 (b1^c)*(b2)^(-1),(b2^c)*(b1*b2)^(-1),
>                 (c^d)*(c^2)^(-1),
>                 (a2^d)*(a1*a2)^(-1),(b2^d)*(b1*b2)^(-1)]));
Group( d, c, a1, a2, b1, b2 )
gap>=20
gap> # now construct Mark's group as a Fp group
gap> fp3 := Group (d,c,a1,a2,b1,b2);
Group( d, c, a1, a2, b1, b2 )
gap> fp3.relators := [b1^2,b2^2,a1^2*(b1)^(-1),a2^2*(b2)^(-1),c^3,d^2,
>                 (a1^c)*(a2)^(-1),(a2^c)*(a1*a2)^(-1),
>                 (b1^c)*(b2)^(-1),(b2^c)*(b1*b2)^(-1),
>                 (c^d)*(c^2)^(-1),
>                 (a2^d)*(a1*a2)^(-1),(b2^d)*(b1*b2)^(-1)];;
gap> IsConsistent (g3);
false    # this is the reason for the trouble
gap> Size (g3); Size (fp3);
96    # so GAP even gets the size wrong
24
```

Note that CharTable (fp3) works without problems, and returns the correct=
result within seconds (on any reasonably fast machine). fp3 is, of course,=
isomorphic with S4.=20

Hope this helps,

Burkhard.

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