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

[Thomas Breuer]

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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