<html><head><title>[xgap] 4.4 A Partial Subgroup Lattice of the Trefoil Knot Group</title></head>
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<h1>4.4 A Partial Subgroup Lattice of the Trefoil Knot Group</h1><p>
<p>
This section investigates the following finitely presented group, the
trefoil knot group <var>K<sub>3</sub></var>.
<p><var>
langlea, b ;;; aba = bab rangle
<p></var>
<p>
This examples shows some limitations of the methods available, in
particular if infinite factors occur.
<p>
<pre>
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">
</pre>
<p>
If you compute the Abelian invariants of <var>K<sub>3</sub></var> 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
<code>Derived Subgroups</code> in the <code>Subgroups</code> menu. A vertex appears in a
level marked with <code>[ infinity, 1 ]</code>. However, there are not too many
things you can do with such infinite index subgroups up to now, as we
will illustrate below:
<p>
First produce some more subgroups by <code>Low Index Subgroups</code> (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, <font face="Gill Sans,Helvetica,Arial">GAP</font> will run into an error:
<p>
<pre>
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)
</pre>
<p>
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.
<p>
You can leave this break loop by entering the command <code>quit;</code> or by
clicking <code>Leave Break Loop</code> in the <code>Run</code> menu of the main XGAP
window.
<p>
Earlier you have computed the subgroups of index at most <var>5</var>. There
is one normal subgroup of index <var>2</var> belonging to vertex <var>6</var> and one of
index <var>4</var> belonging to vertex <var>8</var>. There is <strong>no</strong> line between those
two vertices. Select both and click on <code>Compare Subgroups</code> in the
<code>Subgroups</code> menu. A line appears and the line between vertices <var>8</var> and
<var>G</var> vanishes. The reason for this is, that the
<code>LowIndexSubgroupsFpGroup</code> 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
<code>Compare Subgroups</code>. Note that this can mean large computations, especially if
the indices are huge.
<p>
Now select vertex <var>10</var> and choose <code>Cores</code> from the <code>Subgroups</code> menu.
You will get a new vertex <var>12</var> for an index <var>24</var> subgroup. Select the
vertices <var>12</var> and <var>G</var> and choose <code>Intermediate Subgroups</code> from the
<code>Subgroups</code> 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 <strong>not</strong> performed automatically in the case of finitely
presented groups.
<p>
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
<code>Compare Subgroups</code> from the <code>Subgroups</code> menu. A few vertices will disappear
and you get some messages in the <font face="Gill Sans,Helvetica,Arial">GAP</font> window about merging of
vertices.
<p>
The display is also not fully correct with respect to conjugacy
classes. <code>IntermediateSubgroups</code> does not return the complete
information about conjugacy of subgroups. Because also conjugacy tests
can be very expensive, they are also <strong>not</strong> performed automatically for
finitely presented groups. Select <code>Test Conjugacy</code> from the
<code>Subgroups</code> 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 <var>24</var> above this one you recognize the
subgroup lattice of the symmetric group on <var>4</var> points above that
normal subgroup.
<p>
This is now the end of our partial investigation of the (partial)
subgroup lattice of <var>K<sub>3</sub></var>, close the graphic sheet by selecting <code>close
graphic sheet</code> from the <code>Sheet</code> menu.
<p>
<p>
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<P>
<address>xgap manual<br>Mai 2003
</address></body></html>
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