[GAP Forum] Modified ConjugacyClassesSubgroups?

[Alexander Hulpke]

Dear Forum,

Joe Bohanon asked:
> I have a group G and a subgroup H.  I want to compute the conjugacy  
> classes of subgroups H where conjugacy is taken in G, instead of H.   
> The way I've been doing it is the following:
>
> List(ConjugacyClassesSubgroups(H),Representative);
>
> Then I manually check to see how which classes are fused in G and  
> remove redundacies.
>
> Is there a way to get the same results without doing that?  In other  
> words, could I use the cyclic extension method, except compute the  
> zuppos of H as conjugacy classes of G and do all the required  
> calculations in G instead of H?

You can do so by explicitly calling the cyclic extension method, this  
lets you apply a further filter function. For example:

gap> g:=SymmetricGroup(7);
Sym( [ 1 .. 7 ] )
gap> h:=DerivedSubgroup(g);
Group([ (1,3,2), (2,4,3), (3,4,5), (1,5,6,4,3), (1,3,6,7,5) ])

selector function:

gap> func:=i->IsSubset(h,i);
function( i ) ... end
gap> l:=LatticeByCyclicExtension(g,func);
<subgroup lattice of Sym( [ 1 .. 7 ] ), 37 classes,
3786 subgroups, restricted under further condition l!.func>
gap> c:=ConjugacyClassesSubgroups(l);;
gap> Length(c);
37

# check in a naive way
gap> Length(ConjugacyClassesSubgroups(h));
40
gap> Number(ConjugacyClassesSubgroups(g),i- 
 >IsSubset(h,Representative(i)));
37

Best,

    Alexander Hulpke

-- Colorado State University, Department of Mathematics,
Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
email: hulpke at math.colostate.edu, Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke