Dear GAP-Forum,

"Bjorn Vandenbergh" wrote:

Hello,

I'm a last years student in mathematics and I'm making my thesis. The
name will probably be "Working algebraic with groups: an acquintance
with Gap". In this matter I'm studying the Lattice of subgroups of a
permutation group with Gap. I've read about the theory of lattices of
subgroups. But I cannot interprete the output of the command
LatticeSubgroups(G) with G the permutationgroup.

Here's an example: G is the group of permutations of the triangle in the
plane.

GAP>LatticeSubgroups(G);
<subgroup lattice of Sym( [ 1 .. 3 ] ), 4 classes, 6 subgroups>

I don't understand how I can construct the full lattice when I know only
the number of Conjugacy Classes and the number of Subgroups.

Thank you for helping me

The following simple example shows how you
can get representatives of conjugacy classes
of subgroups of the symmetric group on 3 letters
and learn their orders.

gap> G:= SymmetricGroup(3);
Sym( [ 1 .. 3 ] )
gap> lat := LatticeSubgroups( G );
<subgroup lattice of Sym( [ 1 .. 3 ] ), 4 classes, 6 subgroups>
## It is just a summary of results
gap> KnownAttributesOfObject( G );
[ "Size", "One", "NrMovedPoints", "MovedPoints",
"GeneratorsOfMagmaWithInverses", "TrivialSubmagmaWithOne",
"MultiplicativeNeutralElement", "ConjugacyClasses",
"RepresentativesPerfectSubgroups", "LatticeSubgroups",
"NrConjugacyClasses",
"Pcgs", "BlocksAttr", "StabChainMutable", "StabChainOptions", "Zuppos" ]
## These lines let you know all the attributes
## concerning your group.
gap> cs := ConjugacyClassesSubgroups(G);
[ Group( () )^G, Group( [ (2,3) ] )^G, Group( [ (1,2,3) ] )^G,
SymmetricGroup( [ 1 .. 3 ] )^G ]
gap> sub := List(cs, Representative);
[ Group(()), Group([ (2,3) ]), Group([ (1,2,3) ]), Sym( [ 1 .. 3 ] ) ]

gap> for i in sub do Print(Size(i), ";  "); od; Print("\n");
1;  2;  3;  6;

If you like to get the whole lattice, you should compute
the normalizer N for each subgroup D from the list and
then calculate tr:= RightTransversal(G, N). After that
usage of ConjugateSubgroup(D,x) for each x in tr
allow you to get full lattice. Obviously, in this example
you should add just 2 cyclic subgroups generated by
transpositions (1,2) and (1,3) respectively.
If you have further questions, please contact me directly.

Hope this helps,               Vitaliy Mysovskikh
----------------------------------------------------
Vitaliy Ivanovich Mysovskikh
Associate Prof, PhD
Department of Mathematics and Mechanics
St. Petersburg State University
Bibliotechnaja Pl. 2, St. Petersburg 198904, RUSSIA

E-mail: vimys@pdmi.ras.ru
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