[GAP Forum] Intersections

[Thomas Breuer]

Dear John,

you asked the GAP Forum about computing class representatives
of intersections of sets of k subgroups from given conjugacy classes.

Alexander Hulpke's message from September 26th answers the question
for the case k = 2:
Fix a representative of the first class of subgroups,
and compute its intersection with the members of the second class.
(You may want to reduce the number of candidates using double coset
representatives.)

For k > 2, you can proceed inductively:
Once you know representatives of the intersections of k-1 subgroups,
you can compute their intersections with members of the k-th class.

All the best,
Thomas


On Mon, Oct 12, 2020 at 10:15:39AM +0200, Motalane, John wrote:
> Dear All
> 
> Hoping that you are well.
> 
> I am working with the alternating group A_11. I am looking at the
> intersection of five groups as done in CAP below. I see that the
> intersection lies in these three orbits (120, 360 and 720). I can easily
> calculate (see programme below) that the intersection of  conjugacy classes
> for these five maximal subgroups has candidates of orders 72 and 120. My
> question is that are these the only candidates, if not how do we find other
> candidates? One can show the best method for finding the intersection of
> conjugacy classes for maximal subgroups.
> 
> Thanks in advance
> 
> Regards John
> 
> gap> a:= (1,2,3);;
> gap> b:= (3,4,5,6,7,8,9,10,11);;
> gap> g:=Group(a,b);;
> gap> mx:= MaximalSubgroupClassReps( g );;
> gap> m11:= Filtered( mx, x -> Size(x) = 2^7*3^4 * 175);;
> gap> m22:= Filtered( mx, x -> Size(x) = 2^7*3^4 * 35);;
> gap> m33:= Filtered( mx, x -> Size(x) = 2^7*3^3 * 35);;
> gap> m44:= Filtered( mx, x -> Size(x) = 2^6*3^3 * 35);;
> gap> m55:= Filtered( mx, x -> Size(x) = 2^6*3^3 * 25);;
> gap> m66:= Filtered( mx, x -> Size(x) = 2^4*3^2 * 55);;
> gap> m77:= Filtered( mx, x -> Size(x) = 2^3*3^2 * 110);;
> gap> o:=Orbit(g,m11[1]);;
> gap>
> i12345:=List(o,x->Intersection(x,(Intersection(m22[1],m33[1],m44[1],m55[1]))));;
> gap> List(i12345,Size);
> [ 360, 120, 120, 120, 120, 120, 120, 720, 720, 720, 360 ]
> 
> gap> repeat x:= Intersection( m11[1], m22[1],m33[1],m44[1],m55[1]^Random(g)
> );
> > until Size( x ) <> 1;
> gap> Size(x);
> 72
> gap> repeat x:= Intersection( m11[1], m22[1],m33[1],m44[1],m55[1]^Random(g)
> );
> > until Size( x ) <> 72;
> gap> Size(x);
> 120
> gap> repeat x:= Intersection( m11[1], m22[1],m33[1],m44[1],m55[1]^Random(g)
> );
> > until Size( x ) <> 120;
> gap> Size(x);
> 72