4.4 A Partial Subgroup Lattice of the Trefoil Knot Group

This section investigates the following finitely presented group, the trefoil knot group K₃.

langlea, b ;;; aba = bab rangle

This examples shows some limitations of the methods available, in particular if infinite factors occur.

gap> f := FreeGroup( "a", "b" );
<free group on the generators [ a, b ]>
gap> k3 := f / [ f.1*f.2*f.1 / (f.2*f.1*f.2) ];
<fp group on the generators [ a, b ]>
gap> s := GraphicSubgroupLattice(k3);
<graphic subgroup lattice "GraphicSubgroupLattice">

If you compute the Abelian invariants of K₃ you will see that the commutator factor group is isomorphic to the infinite cyclic group. If you try to compute the derived subgroups it works! Just click on Derived Subgroups in the Subgroups menu. A vertex appears in a level marked with [ infinity, 1 ]. However, there are not too many things you can do with such infinite index subgroups up to now, as we will illustrate below:

First produce some more subgroups by Low Index Subgroups (for example with index limit 5). If you now try to compare one of the new subgroups with the derived subgroup, this is possible. If you however try to calculate the intersection of one of the finite-index subgroups with the derived subgroups, GAP will run into an error:

Error the coset enumeration has defined more than 256000 cosets:
type 'return;' if you want to continue with a new limit of 512000 cosets,
type 'quit;' if you want to quit the coset enumeration,
type 'maxlimit := 0; return;' in order to continue without a limit,
...   (a few lines follow)

This can happen if the coset enumeration algorithm tries to enumerate the cosets of a subgroup with infinite index. This situation can also occur with other operations.

You can leave this break loop by entering the command quit; or by clicking Leave Break Loop in the Run menu of the main XGAP window.

Earlier you have computed the subgroups of index at most 5. There is one normal subgroup of index 2 belonging to vertex 6 and one of index 4 belonging to vertex 8. There is no line between those two vertices. Select both and click on Compare Subgroups in the Subgroups menu. A line appears and the line between vertices 8 and G vanishes. The reason for this is, that the LowIndexSubgroupsFpGroup call did not deliver the complete inclusion info. This can always happen for finitely presented groups in XGAP. In this case you have to compare the subgroups manually by Compare Subgroups. Note that this can mean large computations, especially if the indices are huge.

Now select vertex 10 and choose Cores from the Subgroups menu. You will get a new vertex 12 for an index 24 subgroup. Select the vertices 12 and G and choose Intermediate Subgroups from the Subgroups menu. You will get lots of new vertices. Note that some of them are duplicates of those which were already in the lattice. This is because comparison of subgroups can be quite expensive and is therefore not performed automatically in the case of finitely presented groups.

Select all vertices with a rubber band (click into the top left corner of the sheet, hold down the mouse and move the pointer to the lower right corner, then release the mouse button), and choose Compare Subgroups from the Subgroups menu. A few vertices will disappear and you get some messages in the GAP window about merging of vertices.

The display is also not fully correct with respect to conjugacy classes. IntermediateSubgroups does not return the complete information about conjugacy of subgroups. Because also conjugacy tests can be very expensive, they are also not performed automatically for finitely presented groups. Select Test Conjugacy from the Subgroups menu to trigger this test manually (note that all vertices are still selected!). The vertices belonging to conjugate subgroups are arranged together and if you move those containing the normal subgroup of index 24 above this one you recognize the subgroup lattice of the symmetric group on 4 points above that normal subgroup.

This is now the end of our partial investigation of the (partial) subgroup lattice of K₃, close the graphic sheet by selecting close graphic sheet from the Sheet menu.

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