Dear Forum,
I have a basic question about constructing instransitive G-sets (and
their systems of imprimitivity) in GAP.
I start with a transitive group, say
G := TransitiveGroup(12,8);
(the copy of S4 that acts transitively on 12 points). I compute the
stabilizer of a point, say
H := Stabilizer(G, 1);
Then H = C2, and 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.
I imagine there are a number of ways to do this in GAP, but I have
very little experience using GAP to construct group actions, so any
tips would be greatly appreciated.
Should I be using SparseActionHomorphism? Should I use ExternalSet?
Or should I just use some combination of FactorCosetAction and/or
Action(G,RightTransversal(G,H),OnRight)?
Thanks in advance for any help you can provide!
Sincerely,
William DeMeo
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.
Unfortunately, they don't correspond to the subgroups above some
stabilizer (like we have in the transitive case), but I wonder if the
following conjecture is true: If L is the congruence lattice of an
(intransitive) group action, then there is some \emph{transitive}
group action (presumably coming from a much larger group) with the
same congruence lattice L. If anyone knows that this is true, please
let me know. If you know that it is false, then you have solved the
whole problem [1]. (P.P.S. All groups are finite.)
[1] http://garden.irmacs.sfu.ca/?q=op/finite_congruence_lattice_problem
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