[xgap] 4.3 A Partial Subgroup Lattice of the Cavicchioli Group

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# 4.3 A Partial Subgroup Lattice of the Cavicchioli Group

This section investigates the following finitely presented group C2, which was first investigated by Alberto Cavicchioli in Cav86:

langlea, b ;;; aba-2ba=b, (b-1a3b-1a-3)2=a-1rangle.

In this example we will show a way to prove a finitely presented group to be infinite, and to find some big nonabelian factor groups of it.

The following GAP commands define C2.

    gap> f := FreeGroup( "a", "b" );  a := f.1;;  b := f.2;;
    <free group on the generators [ a, b ]>
    gap> c2 := f / [ a*b*a^-2*b*a/b, (b^-1*a^3*b^-1*a^-3)^2*a ];
    <fp group on the generators [ a, b ]>
    gap> SetName(c2,"c2");

We again 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 and A Partial Subgroup Lattice of the Symmetric Group on 6 Points.

In order to build a partial lattice of a finitely presented group, you again use the function `GraphicSubgroupLattice`. But if the first argument to `GraphicSubgroupLattice` is a finitely presented group the available menus are different from the example in the previous section. After you have entered

    gap> s := GraphicSubgroupLattice(c2);
    <graphic subgroup lattice "GraphicSubgroupLattice of c2">

XGAP will open a window containing a new graphic sheet. Compared to the interactive lattice of a permutation group as described in the previous section, there are the following differences:

-- There is only one vertex instead of two. This vertex labeled G is the whole group C2. There is no vertex for the trivial subgroup (yet).

-- If you pull down the `Subgroups` menu, you will see that this menu is now very different. It gives you access to various algorithms for finitely presented groups but most of the entries from the last two examples are missing because most of the GAP functions behind these entries are not applicable to (infinite) finitely presented groups.

This example will show you how to prove that C2 is infinite. First look at the abelian invariants in order to see what the commutator factor group is. In order to compute the abelian invariants pop up the ``Information'' menu. This is done in exactly the same manner as in the previous section. Place the pointer inside vertex G, press the right mouse button and release it immediately. This ``Information'' menu is described in detail in GraphicSubgroupLattice for FpGroups, Information Menu.

    Index              1
    IsNormal           true
    IsFpGroup          unknown
    Abelian Invariants unknown
    Coset Table        unknown
    IsomorphismFpGroup unknown
    Factor Group       unknown

This tells you what XGAP already knows about the group associated with vertex G. In order to compute the abelian invariants click onto this line. After a while this entry will change to

    Abelian Invariants perfect

telling you that C2 is perfect. So none of the `Subgroups` menu entries `Abelian Prime Quotient`, `All Overgroups`, `Conjugacy Class`, `Cores`, `Derived Subgroups`, `Intersection`, `Intersections`, `Normalizers` or `Prime Quotient` will compute any new subgroups.

In order to avoid accidents the menu entries `Abelian Prime Quotient`, `All Overgroups`, `Epimorphisms (GQuotients)`, `Conjugacy Class`, `Low Index Subgroups`, and `Prime Quotient` from the `Subgroups` menu are only selectable if exactly one vertex is selected because the functions behind these entries are in general quite time and space consuming.

Close the ``Information'' window and select `Low Index Subgroups` from the `Subgroups` menu. A small dialog box will pop up asking for a limit on the index. Type in 12 and press return or click on `OK`. In general it is hard to say what kind of index limit will still work, for some groups even 5 might be too much while for others 20 works fine, see also LowIndexSubgroupsFpGroup.

GAP computes 10 subgroups of index 11 and 8 subgroups of index 12. If you now start to check the abelian invariants of the index 12 subgroups you will find out that all subgroups represented by vertices 3 to 10 have a finite commutator factor group except the subgroup belonging to vertex 4 which has an infinite abelian quotient. Therefore the group C2 itself is infinite.

Now we want to investigate C2 a little further using GAP. Select vertices 3, 4, and 5 and switch to the GAP window. Use `SelectedGroups` to get the subgroups associated with these vertices.

    gap> u := SelectedGroups( s );
    [ Group([ a, b*a^2*b^-2, b*a*b^2*a^-1*b^-1*a^-1*b^-1, b^4*a^-2*b^-2,
          b^2*a^3*b^-1*a^-1*b^-2 ]),
      Group([ a, b^2*a*b^-1*a^-1*b^-1, b^3*a^-1*b^-1, b*a*b*a^3*b^-1 ]),
      Group([ a, b^2*a*b^-1*a^-1*b^-1, b*a^3*b^-2, b^4*a^-1*b^-3,
          b*a*b^3*a^-1*b^-1 ]) ]

`FactorCosetOperation` computes for each of these subgroups ui the operation of C2 on its cosets. It returns the result as a homomorphism of C2 onto a permutation group. The operation on ui is therefore a permutation representation of the factor group

C2 / Core(ui).

Using `DisplayCompositionSeries` we can identify these factor groups.

    gap> p := List( u, x -> FactorCosetOperation( c2, x ) );;
    gap> l := List( p, Image );;
    gap> for x  in l  do DisplayCompositionSeries(x);  Print("\n");  od;
    G (2 gens, size 95040)
     | M(12)
    1 (0 gens, size 1)

    G (2 gens, size 660)
     | A(1,11) = L(2,11) ~ B(1,11) = O(3,11) ~ C(1,11) = S(2,11) ~ 2A(1,11) = U(2,11)
    1 (0 gens, size 1)

    G (2 gens, size 239500800)
     | A(12)
    1 (0 gens, size 1)

(This display can look a little different according to the GAP version you use.)

So C2 contains the Mathieu group M12, the alternating group on 12 symbols and PSL(2,11) as factor groups. Therefore it would have been possible to find vertex 4 using `Epimorphisms (GQuotients)` instead of `Low Index Subgroups`. Close the graphic sheet by selecting the menu entry `close graphic sheet` from the `Sheet` menu and start with a fresh one.

    gap> s := GraphicSubgroupLattice(c2);
    <graphic subgroup lattice "GraphicSubgroupLattice of c2">

Select `Epimorphisms (GQuotients)` from the `Subgroups` menu. This pops up a menu similar to the ``Information'' menu (see GraphicSubgroupLattice for FpGroups, Subgroups Menu).

    Sym(n)
    Alt(n)
    PSL(d,q)
    Library
    User Defined

Select PSL(d,q), which pops up a dialog box asking for a dimension. Enter `2` and click on OK. Then a second dialog box pops up asking for a field size. Enter `11` and click on OK. After a short time of computation the display in the `Epimorphisms (GQuotients)` menu changes and shows

    PSL(2,11)      1 found

telling you, that GAP has found 1 epimorphism (up to inner automorphisms of PSL(2,11)) from C2 onto PSL(2,11). Click on display point stabilizer to create a new vertex representing a subgroup u such that the factor group of C2 / Core(u) is isomorphic to PSL(2,11). You could have clicked on display to create a new vertex representing the kernel of the epimorphism.

This is now the end of our partial investigation of the (partial) subgroup lattice of C2, you have seen that C2 is infinite and contains M12, Alt(12), and PSL(2,11) as factor groups. Close the graphic sheet by selecting `close graphic sheet` from the `Sheet` menu.

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