[xgap] 4.5 A Partial Subgroup Lattice of a Finitely Presented Group

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# 4.5 A Partial Subgroup Lattice of a Finitely Presented Group

This section describes the investigation of the following finitely presented group:

G := langlea, b ; ; ; a6 = 1 rangle

We will show especially how to deal with subgroups which have a very large index like those occuring in the prime quotient algorithm.

Define the group and open the subgroup lattice window:

    gap> f := FreeGroup(2);
    <free group on the generators [ f1, f2 ]>
    gap> g := f/[f.1^6];
    <fp group on the generators [ f1, f2 ]>
    gap> s := GraphicSubgroupLattice(g);
    <graphic subgroup lattice "GraphicSubgroupLattice">

First compute prime quotients by `Prime Quotient` in the `Subgroups` menu. You are asked for a prime and a class. Enter 2 and 7 respectively. You get lots of output in the GAP command window and seven new vertices. Some of the corresponding subgroups have huge indices. Note that these groups are only represented as kernels of epimorphisms within GAP. So explicit calculation of a coset table or a presentation could take very long or be absolutely impossible!

Now compute epimorphisms onto the symmetric group on 3 points by `Epimorphisms (GQuotients)` in the `Subgroups` menu, but use a polycyclic presentation as follows (the reason for this will be explained below):

    gap> IdGroup(SymmetricGroup(3));
    [ 6, 1 ]
    gap> s3 := SmallGroup(6,1);
    <pc group with 2 generators>
    gap> IMAGE_GROUP := s3;;

This first determines the identification number of the symmetric group on 3 points within the small groups library, and then fetches this group as a polycyclic group. For groups of size less than 1000 this is often a good way to get a polycyclic presentation. Note that `SymmetricGroup(3)` leads to a permutation group. The last statement stores the group in a variable which can be used by XGAP.

Select vertex G, then click on `Epimorphisms (GQuotient)` in the `Subgroups` menu and select `User defined` in the window that pops up. This will always use the group stored in the global variable `IMAGE_GROUP`. GAP finds three epimorphisms. Display the three kernels by selecting display in the epimorphisms window.

Note that the new vertex 9 will be drawn on the line between vertices 2 and 3 because there is not yet a vertex in the level corresponding to index 6. You can move it aside by dragging it with the mouse to some better position within its level.

Now select vertices 8 and 11 and calculate the intersection of the two subgroups of indices 137438953472 and 6 respectively. GAP can calculate this intersection by calculating the subdirect product of the image groups of the epimorphisms (the index of the subgroup belonging to vertex 12 in G is 412316860416 which is three times of the index of the subgroup belonging to vertex 11). Note that this subdirect product can only be calculated because the two image groups are polycyclic groups. This is the reason why we needed S3 as polycyclic group earlier.

This is now the end of our partial investigation of the (partial) subgroup lattice of G, close the graphic sheet by selecting `close graphic sheet` from the `Sheet` menu.

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