[GAP Forum] Forum Digest, Vol 194, Issue 7

[Siddhartha Sarkar]

Dear Thomas,

As you already mentioned that the natural construction through matrices in
GAP swaps
the classes 7A, 7B from what is standardised Atlas nomenclature.

For small number of classes this is easily detected and while computing one
can rectify.
Are these changes random? i.e., each time I run it in GAP should it permute
the classes
randomly?

If yes, is there a way to detect this and rectify in the subroutine while
writing a program?

Best regards,
Siddhartha

Message: 1
> Date: Wed, 29 Jan 2020 20:02:30 +0100
> From: Thomas Breuer <sam at Math.RWTH-Aachen.De>
> To: GAP Forum <forum at gap-system.org>
> Subject: Re: [GAP Forum] Computing conjugacy classes on matrix groups
>         over finite fields
> Message-ID:
>         <20200129190230.yvstwqq6m4k56ocj at hamal.math.rwth-aachen.de>
> Content-Type: text/plain; charset=iso-8859-1
>
> Dear Forum,
>
> since Siddharta asked about conjugay of elements in a group,
> without mentioning the context of character tables,
> I would propose just to use 'IsConjugate'.
>
>     gap> x:= [ [1,1,0], [0,1,0], [0,0,1] ] * Z(2);;
>     gap> y:= [ [0,0,1], [1,0,0], [0,1,0] ] * Z(2);;
>     gap> g:= Group( x, y );;
>     gap> Size( g );
>     168
>     gap> xy:= x * y;;
>     gap> Order( xy );
>     7
>     gap> ccl:= ConjugacyClasses( g );;
>     gap> reps:= List( ccl, Representative );;
>     gap> List( reps, Order );
>     [ 1, 2, 4, 7, 7, 3 ]
>     gap> xy in ccl[4];
>     true
>     gap> xy in ccl[5];
>     false
>
> I think that it makes no sense to ask in which of the two classes
> the element 'xy' lies, unless one has defined what ``the first of the
> two classes'' means.
> Note that there is a group automorphism that swaps the two classes.
> But once you have fixed a class, you can ask whether 'xy' lies in that
> class.
>
>     gap> IsConjugate( g, xy, reps[4] );
>     true
>
> And it makes sense to ask whether some power of 'xy' is conjugate to 'xy'.
>
>     gap> List( [ 1 .. 6 ], i -> IsConjugate( g, xy^i, xy ) );
>     [ true, true, false, true, false, false ]
>
> All the best,
> Thomas
>
>