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 <ahul...@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: hul...@math.colostate.edu, Phone: ++1-970-4914288 > http://www.math.colostate.edu/~hulpke > > > > _______________________________________________ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum
