[GAP Forum] intransitive G-sets

[Alexander Hulpke]

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,
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