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

[Siddhartha Sarkar]

Thanks Thomas,

I see it now. I somehow always thought that the letters on the classes is
somewhat named in certain way
and came through Atlas. I was wrong about it.

For the fixed point (of actions of Riemann surfaces) calculations, I am
trying to verify : for L_3(2) the characters
X_2 and X_3 are symmetric via choices of generators. So in this setting the
naming 7A and 7B doesn't play much
role. In general, I need to be careful perhaps.

Best regards,
Siddhartha

On Sun, Feb 9, 2020 at 10:17 PM Thomas Breuer <sam at math.rwth-aachen.de>
wrote:

> Dear Siddharta,
>
> thank you for your reply.
>
> I do not understand your first statement.
> The group GL(3,2) has two conjugacy classes of elements of order 7,
> but the ATLAS of Finite Groups does not define which of them is called 7A.
> Defining class names for an abstract group makes sense only up to group
> automorphisms, and the group automorphism that maps each meatrix in GL(3,2)
> to its transposed inverse swaps the two classes of element order 7.
>
> Concerning the GAP function 'ConjugacyClasses', in general the ordering
> of the result list can be different for the same group in different GAP
> sessions, or for different group objects in the same GAP session
> which happen to be equal as sets.
> (However, the class of the identity element always comes first.)
> There are groups for which in fact such differences cannot be observed
> with the current GAP, but you cannot rely on the fact that this will
> still be the case with the next version of GAP.
>
> All the best,
> Thomas
>
>
> On Thu, Feb 06, 2020 at 07:56:25PM +0530, Siddhartha Sarkar wrote:
> > 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
> > >
> > >
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