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 <s...@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 <s...@math.rwth-aachen.de> > > > To: GAP Forum <forum@gap-system.org> > > > Subject: Re: [GAP Forum] Computing conjugacy classes on matrix groups > > > over finite fields > > > Message-ID: > > > <20200129190230.yvstwqq6m4k56...@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 > > > > > > > > _______________________________________________ > > Forum mailing list > > Forum@gap-system.org > > https://mail.gap-system.org/mailman/listinfo/forum > _______________________________________________ Forum mailing list Forum@gap-system.org https://mail.gap-system.org/mailman/listinfo/forum
