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On Sat, May 19, 2001 at 01:28:03PM +0800, MICHAEL HARTLEY wrote:

> In section 36.8 of the GAP4 manual says that "For element

> operations such as in, a double coset behaves like a set of group

> elements".

Hi Michael,

I guess the problem here is that there is no `Elements' method for

double cosets. I'm not sure whether this is intentional; perhaps someone

else will follow up with some more details. We could have done with an

example here of how you intended to use these functions. If you wanted

to print out the elements, then I guess your groups are small and so

the following strategy should do you:

There is an `Elements' method for right cosets and right transversals.

So for a double coset HgK rewrite as:

-1 -1 HgKg g = U Htg, where T = RightTransversal(gKg , L) and t in T -1) L = Intersection(H, gKg )

i.e. using this approach the double coset HgK is realised as a union

of right cosets. In fact, GAP4 provides: `RepresentativesContainedRightCosets'

which essentially returns the set Tg. So here's an example of how one can

do it:

gap> G := Group([ (1,2,3,4), (1,2) ]); Group([ (1,2,3,4), (1,2) ]) gap> H := Subgroup(G, [(1,2,3), (1,2)]); Group([ (1,2,3), (1,2) ]) gap> K := Subgroup(G, [(3,4)]); Group([ (3,4) ]) gap> g := (2,4); (2,4) gap> D := DoubleCoset(H, g, K); DoubleCoset(Group( [ (1,2,3), (1,2) ] ),(2,4),Group( [ (3,4) ] )) gap> Tg := RepresentativesContainedRightCosets(D); [ (2,3,4) ] gap> D_as_right_cosets := Union(List(Tg, tg -> RightCoset(H, tg))); RightCoset(Group( [ (1,2,3), (1,2) ] ),(2,3,4)) gap> Elements(D_as_right_cosets); [ (2,3,4), (2,4), (1,3,4,2), (1,3)(2,4), (1,4,2), (1,4,2,3) ]

If this is the right strategy for you, you could define a function:

DoubleCosetAsRightCosets := function(H, g, K)

local D, Tg;

D := DoubleCoset(H, g, K);

Tg := RepresentativesContainedRightCosets(D);

return Union(List(Tg, tg -> RightCoset(H, tg)));

end;

Then just use `DoubleCosetAsRightCosets' where you would otherwise

have used `DoubleCoset'.

On the other hand, if your groups are big, perhaps you don't really

want to do this at all. In this case, you probably only want to check

whether certain elements are in the intersection or union and in such

cases you should probably translate these to the corresponding logical

`and' and `or' operations i.e.

x in Intersection(H, K) <=> x in H and x in K x in Union(H,K) <=> x in H or x in K

and in this case `DoubleCoset' will be the right function for you to

use.

So, I guess the question is: `How big are your groups?'

But when I try to take the union or intersection of a DoubleCoset

with a Set of group elements, GAP4 can't do it! It enters the break

loop...How can I (easily) take these unions and intersections??

I hope one of the above solutions is the right one for you and satisfies

the `easy' criterion.

Regards, Greg Gamble ___________________________________________________________________ Greg Gamble __________________ mailto:gregg@math.rwth-aachen.de Lehrstuhl D fuer Mathematik Tel: +49 241 804545 Templergraben 64 52062 Aachen, Germany http://www.math.rwth-aachen.de/~Greg.Gamble ___________________________________________________________________

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