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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