> < ^ Date: Fri, 24 Nov 2000 17:23:01 -0000
> < ^ From: Kurt Ewald <Kurt.Ewald@balbec.de >
^ Subject: AW: Subgroup Lattice??

gap forum gap-forum.

Hallo,

if you want a short overwiew of the Subgroups - f.i. s4 -
then use the little program

L:=[];
Net:=function(lat,m)
# shows an overview of the lattice
# lat: name of LatticeSubgroups, a: number of the ConjugacyClasses
local i;
for i in [1..m] do
L:=List(ConjugacyClassesSubgroups(lat)[i],Size);Print(L,"
",Length(L),"\n");
od;
end;

```gap> s4:=SymmetricGroup(4);
Sym( [ 1 .. 4 ] )
gap> lat:=LatticeSubgroups(s4);
<subgroup lattice of Sym( [ 1 .. 4 ] ), 11 classes, 30 subgroups>
gap> Net(lat,11);
[ 1 ] 1
[ 2, 2, 2 ] 3
[ 2, 2, 2, 2, 2, 2 ] 6
[ 3, 3, 3, 3 ] 4
[ 4 ] 1
[ 4, 4, 4 ] 3
[ 4, 4, 4 ] 3
[ 6, 6, 6, 6 ] 4
[ 8, 8, 8 ] 3
[ 12 ] 1
[ 24 ] 1
```

If you want to see a distinct Subgroup use the following
NetSubgroup:=function(lat,a,b)
# shows a distinct Subgroup
# lat: name of LatticeSubgroups, a: number of the wanted ConjugacyClass,
# b: number of the wanted Subgroup
return ConjugacyClassesSubgroups(lat)[a][b];
end;

```gap> NetSubgroup(lat,9,2);
Group([ (2,4), (1,3), (1,4)(2,3) ])
```

if you want to see all the groups of ConjugacyClass then use

NetSubgroups:=function(lat,a,n1,n2)
# lat: name of LatticeSubgroups, a: number of the wanted ConjugacyClass
# n1: first Subgroup, n2: last Subgroup of the ConjugacyClass
local i;
for i in [n1..n2] do Print (ConjugacyClassesSubgroups(lat)[a][i], "\n");od;
end;

```gap> NetSubgroups(lat,9,1,3);
Group( [ (3,4), (1,2), (1,3)(2,4) ] )
Group( [ (2,4), (1,3), (1,4)(2,3) ] )
Group( [ (2,3), (1,4), (1,2)(3,4) ] )
```

Best wishes

K.Ewald

-----Ursprüngliche Nachricht-----
Von: GAP-Forum-Sender@dcs.st-and.ac.uk
[mailto:GAP-Forum-Sender@dcs.st-and.ac.uk]Im Auftrag von Bjorn
Vandenbergh
Gesendet: Donnerstag, 23. November 2000 16:36
An: Multiple recipients of list
Betreff: Subgroup Lattice??

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

-----Ursprüngliche Nachricht-----
Von: GAP-Forum-Sender@dcs.st-and.ac.uk
[mailto:GAP-Forum-Sender@dcs.st-and.ac.uk]Im Auftrag von Bjorn
Vandenbergh
Gesendet: Donnerstag, 23. November 2000 16:36
An: Multiple recipients of list
Betreff: Subgroup Lattice??

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

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