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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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