[xgap] 4.2 A Partial Subgroup Lattice of the Symmetric Group on 6 Points

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# 4.2 A Partial Subgroup Lattice of the Symmetric Group on 6 Points

This section investigates the subgroup lattice of S6.

    gap> s6 := SymmetricGroup(6);
    Sym( [ 1 .. 6 ] )
    gap> SetName(s6,"S6");
    gap> cc := ConjugacyClassesSubgroups(s6);;
    gap> Sum(List(cc,Size));
    1455

As there are 1455 subgroups, displaying the whole lattice of subgroups would not be helpful because there are simply too many. Therefore this example builds only a partial subgroup lattice. We assume that you are familiar with the general ideas, mouse actions and menus, which were discussed in The Subgroup Lattice of the Dihedral Group of Order 8.

We again start to build a partial lattice, by using `GraphicSubgroupLattice` (see GraphicSubgroupLattice). After you have entered

    gap> s := GraphicSubgroupLattice(s6);
    <graphic subgroup lattice "GraphicSubgroupLattice of S6">

XGAP will open a window containing a new graphic sheet with two connected vertices labeled 1 and G. Vertex 1 represents the trivial subgroup and vertex G the group S6. Vertex G is already selected, so it will be red if your screen supports color.

XGAP can automatically write a protocol of all the subsequent actions you perform via mouse clicks. This is convenient because in comparison to normal GAP sessions you do not have the script of typed commands. You can activate this feature by selecting `Start Logging` from the `Subgroups` menu. XGAP prompts you for a filename via a file selector box. See loggingfacility for details about this feature.

In order to find all subgroups of size 60, we cannot not use the `Subgroups` menu directly, so go back into the GAP window and extract the conjugacy classes of `cc` whose representatives have size 60.

    gap> c60 := Filtered(cc,x->Size(Representative(x))=60);;
    gap> s60 := List(c60,Representative);
    [ Group([ (1,2)(3,4), (1,3,5) ]), Group([ (1,2)(3,4), (1,2,3)(4,5,6) ]) ]

labelxgapgap We now use the function `InsertVertex` (see InsertVertex) to add these two subgroups to your partial lattice.

    gap> for g in s60 do InsertVertex(s,g); od;

Note that we could have achieved this result with the entry `InsertVertices from GAP` in the `Subgroups` menu, see InsertVertex. The Hasse diagram now contains four vertices. The new vertices are not selected automatically. You can do this as mentioned above by clicking with the left mouse button on them. Selecting `Conjugate Subgroups` from the `Subgroups` menu adds the complete conjugacy classes. Please do this first for vertex 2 and then for vertex 3 such that the numbering of vertices in the following description is correct!

In order to find out what type of subgroups we are looking at, use another kind of menu not discussed so far, namely the ``Information'' menu. Place the pointer inside vertex 3, press the right mouse button and release it immediately. This will pop up a new window, containing some text describing vertex 3 (as mentioned above, depending on the window system and window manager, placing this window on the screen might require some interaction with the mouse).

    Size        60
    Index       12
    IsAbelian   unknown
    IsCentral   unknown
    IsCyclic    unknown
    IsNilpotent false
    IsNormal    false
    IsPerfect   true
    IsSimple    unknown
    IsSolvable  false
    Isomorphism unknown

Note that GAP does not yet automatically draw the conclusion that a nonsolvable subgroup is also not abelian, cyclic or central. Place the pointer on top of the entry ``Isomorphism'' and press the left mouse button. After a while this entry is changed to

    Isomorphism [ 60, 5 ]

telling you that the subgroup represented by vertex 3 is isomorphic to the alternating group on five symbols. The notation `[ 60, 5 ]` comes out of the small groups library and is the only information about the isomorphism type we can get from GAP4 up to now. Select close to close the ``Information'' menu. Repeat this with vertex 2, you will see that the subgroup of vertex 2 is also isomorphic to A5, however these two A5 inside S6 are not conjugate in S6. The ``Information'' menu is described in detail in GraphicSubgroupLattice, Information Menu.

Now we want to compute the normalizers of the elements of the conjugacy class containing the subgroup of vertex 3. You could either select vertex 3 and then Normalizers and repeat this process for the vertices 9 to 13, or you can first select the vertices 3, 9 to 13 and then select Normalizers. But how to select more than one vertex? If you first select 3 and then 9, vertex 3 will get deselected as soon as 9 gets selected. However, if you select vertex 3, place the pointer inside vertex 9, hold down the SHIFT key on your keyboard and then select vertex 9 using the left mouse button, vertex 9 will be selected in addition to vertex 3. Another method to select more than one vertex is to use the rubber band to catch vertices inside a rectangle. Place the pointer left and a bit higher than vertex 3 outside any other vertex. Press the left mouse button and hold it down. Now, using the mouse, move the pointer right and slightly below vertex 13. You see a rectangle, one corner at your start position and the other following the pointer. If vertices 3 and 9 to 13 are all inside this rectangle, release the mouse button. Now these vertices are selected. Select `Normalizers` from the `Subgroups` menu to compute and display the normalizers.

Now select vertex 3 and 4 and compute the intersection. The intersection is of size 10. Select this intersection and use `SelectedGroups` to get a GAP record describing the subgroup.

    gap> l := SelectedGroups(s);
    [ Group([ (2,3)(4,6), (1,2)(3,4) ]) ]
    gap> u := l[1];
    Group([ (2,3)(4,6), (1,2)(3,4) ])

In order to find out which subgroups of the complete lattice lie above the subgroup `u` you can use `Intermediate Subgroups`. You select the whole group in addition to `u` and choose `Intermediate Subgroups` in the `Subgroups` menu. You get 6 groups, some of them are already in the lattice, the others are added.

There is another feature we have not seen yet. Close the current graphic sheet and start again with a fresh one.

    gap> Close(s);
    gap> s := GraphicSubgroupLattice(s6);
    <graphic subgroup lattice "GraphicSubgroupLattice of S6">

In order to compute a Sylow 2 subgroup select `Sylow Subgroup` from the `Subgroups` menu. A small dialog box will pop up asking for a prime, type in 2 and press return or click on `OK`. Now select this new vertex 2 representing the Sylow 2 subgroup and compute its normal subgroups. This is rather slow because the function checks for each new vertex if the corresponding subgroup is conjugate to an old one of the same size.

This is now the end of our partial investigation of the (partial) subgroup lattice of S6, close the graphic sheet(s) using `close graphic sheet` of the `Sheet` menu. If you started the logging facility of XGAP as described above you now have a file (probably called `xgap.log` if you did not change the default) describing the actions we performed.

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