[GAP Forum] ConjugacyClasses

[Stefan Kohl]

On Thu, February 7, 2013 3:34 pm, Stefan Witzel wrote:
> can anyone explain the following behavior to me (saying that PSL(3,2) has to distinct
> conjugacy classes that are conjugate)?
>
> vec1:=[1,0,0]*Z(2)^0;;
> sl:=SL(3,2);;
> orb:=Orbit(sl,vec1,OnLines);;
> act:=ActionHomomorphism(sl,orb,OnLines);;
> psl:=Image(act);;
> ConjugacyClasses(psl);
>
> [ ()^G, (3,4)(6,7)^G, (2,3,5,4)(6,7)^G, (2,3,6)(4,7,5)^G, (1,2,3,4,7,5,6)^G,
> (1,2,3,5,6,7,4)^G ]
>
> g:=Representative(ConjugacyClasses(psl)[5]);;
> h:=Representative(ConjugacyClasses(psl)[6]);;
> c:=(2,6,4)(3,5,7);;
> c*g^(-1)*c^(-1)=h;
>
> true
>
> IsSubgroup(psl,Group(c));
>
> true

The point is that the 6th conjugacy class in the list consists of the
inverses of the elements of the 5th conjugacy class (in particular the
elements in these classes are not conjugate to their inverses):

gap> ccl := ConjugacyClasses(psl);
[ ()^G, (3,4)(6,7)^G, (2,3,5,4)(6,7)^G, (2,3,6)(4,7,5)^G, (1,2,3,4,7,5,6)^G,
  (1,2,3,5,6,7,4)^G ]
gap> Set(AsList(ccl[6])) = Set(List(AsList(ccl[5]),g->g^-1));
true
gap> IsConjugate(psl,Representative(ccl[5]),Representative(ccl[6]));
false
gap> IsConjugate(psl,Representative(ccl[5]),Representative(ccl[6])^-1);
true

Hope this helps,

    Stefan Kohl