[GAP Forum] intransitive G-sets

[William DeMeo]

Dear Dr. Hulpke,

Thank you again for another excellent answer.  It is all very clear,
and very helpful.  I just have a comment/question about your caveat.

You wrote:
> Let me add a caveat here: I believe the imprimitivity tests in GAP aleways
> assume that the group acts transitively. So you would have to determine
> congruences for each orbit separately and then combine.

When you say "and then combine," did you have a strategy in mind, or
did you just mean "and then you'll have to figure out how to combine"?

This gets at the heart a basic question we're thinking about.  When
you combine, you should get a direct product of the congruences of the
nontrivial subalgebras (orbits), and a partition lattice above that,
plus some other stuff.  The question is, what is the other stuff?
(Perhaps with your deep knowledge of this area, you already know the
answer; i.e. how to "combine" the block systems of the transitive
subalgebras of an intransitive G-set.)

Since GAP assumes transitivity when computing block systems, perhaps
the best strategy for now is to use GAP to compute an operation table
for the intransitive G-set, and then import this table into the
universal algebra calculator (www.uacalc.org), which can compute the
congruences.  I'll try it today.

Maybe future versions of GAP can include a Blocks function which works
for intransitive actions -- or, better yet, a Con function for
computing (and drawing with XGAP?) the lattice of all block systems of
an intransitive G-set.  I'd be happy to contribute to this effort, if
you think people (besides me) might find it useful.

Thanks again for all your help!!

-William


On Sun, Mar 20, 2011 at 12:59 PM, Alexander Hulpke <ahulpke at gmail.com> wrote:
> Dear GAP Forum,
>
> William DeMeo asked:
>>  G acts transitively on the 12 cosets G/H, (and the
>> systems of imprimitivity correspond to the subgroups of G above H).
>>
>> Now, with the same group G, take another subgroup, K < G, and consider
>> the action on the cosets G/K.  I would like to form the (intransitive)
>> group action on the set G/H union G/K.  That is, I now have two
>> orbits, the original 12 cosets of H, and the cosets of K.
>
> In principle, GAP can calculate the permutation action on any intransitive domain. Thus, for example, if you take the disjoint union of cosets, you could simply calculate the permutation action:
>
> gap> G:=SymmetricGroup(4);
> Sym( [ 1 .. 4 ] )
> gap> H:=Subgroup(G,[(1,2)]);
> Group([ (1,2) ])
> gap> K:=Subgroup(G,[(1,2,3,4)]);
> Group([ (1,2,3,4) ])
> gap> dom:=Concatenation(RightCosets(G,H),RightCosets(G,K));
> [ RightCoset(Group( [ (1,2) ] ),()), RightCoset(Group( [ (1,2) ] ),(2,4)),
>  RightCoset(Group( [ (1,2) ] ),(1,2,4)), RightCoset(Group( [ (1,2) ] ),(1,3)),
>  ....
> gap> act:=Action(G,dom,OnRight);
> Group([ (1,10,5,9)(2,12,4,8)(3,11,6,7)(14,17,16,15),
>  (2,3)(4,7)(5,8)(6,9)(10,11)(13,16)(14,18)(15,17) ])
> gap> Orbits(act,MovedPoints(act));
> [ [ 1, 10, 5, 11, 9, 8, 6, 2, 7, 12, 3, 4 ], [ 13, 16, 15, 14, 17, 18 ] ]
>
> Of course, when constructing a transitive action a convenient (and memory-saving!) shorthand is to act on the RightTransversal by right multiplication. This does not work here, as transversals are special objects and this speciality is destroyed by taking a union or concatenation. In this situation one would have to construct the transitive permutatioon actions (same as `FactorCosetAction') first, and then embed in the direct product.
>
>>
>> P.S.
>> More background on my problem in case you're curious (but feel free to ignore):
>>
>> Ultimately, I want to look at the lattice of all systems of
>> imprimitivity (i.e. congruences) of such intransitive G-sets.
> Let me add a caveat here: I believe the imprimitivity tests in GAP aleways assume that the group acts transitively. So you would have to determine congruences for each orbit separately and then combine.
>
> Hope this helps,
>
>   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
>
>
>
>