[GAP Forum] Find a preserved partition given an imprimitive group action

[William DeMeo]

On Sun, Jun 10, 2012 at 7:19 AM, Nathann Cohen <nathann.cohen at gmail.com> wrote:
> This is just *TOTALLY* great ! :-)
>
> Thank you sooooooooo much !! I think I will spend yet another night playing
> with GAP :-)
>
> Nathann

You are very welcome.

By the way, when the congruence lattice gets big, you may not want to
look at the congruences in the format I gave above as it is sometimes
tedious to figure out the partial ordering of the congruences by hand.
 In that case, you could use the theorem that says the congruence
lattice is isomorphic to the lattice of subgroups above a stabilizer
subgroup, and you can find the latter using GAP in a number of ways.
One way is

gap> H:=Stabilizer(G,1);  # take the stabilizer of any point, e.g., 1.
gap> IntermediateSubgroups(G,H);

The second command tells you all the subgroups of G that contain H,
and also gives the covering relations in the subgroup lattice.

In the dihedral example from my last email (which is regular, so H=1), you get

inclusions := [ [ 0, 1 ], [ 0, 2 ], [ 0, 3 ], [ 0, 4 ], [ 0, 5 ], [ 1,
6 ], [ 2, 6 ], [ 3, 6 ], [ 1, 7 ], [ 1, 8 ], [ 4, 8 ],
      [ 5, 8 ], [ 6, 9 ], [ 7, 9 ], [ 8, 9 ] ] )

As you can see, there are five minimal congruences, or atoms
(corresponding to the labels 1, 2, ..., 5) which cover 0, and three
maximal congruences, or co-atoms (corresponding to 6,7,8) covered by
9.

You can also draw the Hasse diagram of the subgroup lattice if you
have the xgap package installed.

Alternatively, if you really want to study congruence lattices in
detail, you may want to try my script gap2uacalc.g for converting GAP
groups and G-sets into universal algebras that can be imported into
the Universal Algebra Calculator; see www.uacalc.org.

-William





-- 
William J. DeMeo, Ph.D.
Department of Mathematics
University of Hawaii at Manoa
phone: 808-298-4874
url: http://math.hawaii.edu/~williamdemeo