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<html><head><title>[xgap] 4.5 A Partial Subgroup Lattice of a Finitely Presented Group</title></head>
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<h1>4.5 A Partial Subgroup Lattice of a Finitely Presented Group</h1><p>
<p>
This section describes the investigation of the following finitely presented
group: 
<p><var> G := langlea, b ; ; ; a<sup>6</sup> = 1 rangle<p></var>
We will show especially how to deal with subgroups which have a very large
index like those occuring in the prime quotient algorithm.
<p>
Define the group and open the subgroup lattice window:
<p>
<pre>
gap&gt; f := FreeGroup(2);
&lt;free group on the generators [ f1, f2 ]&gt;
gap&gt; g := f/[f.1^6];
&lt;fp group on the generators [ f1, f2 ]&gt;
gap&gt; s := GraphicSubgroupLattice(g);
&lt;graphic subgroup lattice "GraphicSubgroupLattice"&gt;
</pre>
<p>
First compute prime quotients by <code>Prime Quotient</code> in the <code>Subgroups</code>
menu. You are asked for a prime and a class. Enter <var>2</var> and <var>7</var>
respectively. You get lots of output in the <font face="Gill Sans,Helvetica,Arial">GAP</font> 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 <font face="Gill Sans,Helvetica,Arial">GAP</font>. So explicit calculation of a coset table
or a presentation could take very long or be absolutely impossible!
<p>
Now compute epimorphisms onto the symmetric group on <var>3</var> points by
<code>Epimorphisms (GQuotients)</code> in the <code>Subgroups</code> menu, but use a polycyclic 
presentation as follows (the reason for this will be explained below):
<p>
<pre>
gap&gt; IdGroup(SymmetricGroup(3));
[ 6, 1 ]
gap&gt; s3 := SmallGroup(6,1);
&lt;pc group with 2 generators&gt;
gap&gt; IMAGE_GROUP := s3;;
</pre>
<p>
This first determines the identification number of the symmetric group 
on <var>3</var> points within the small groups library, and then fetches this
group as a polycyclic group. For groups of size less than <var>1000</var> this
is often a good way to get a polycyclic presentation. Note that
<code>SymmetricGroup(3)</code> leads to a permutation group. The last statement
stores the group in a variable which can be used by XGAP. 
<p>
Select vertex <var>G</var>, then click on <code>Epimorphisms (GQuotient)</code> in the
<code>Subgroups</code> menu and select <code>User defined</code> in the window that pops up.
This will always use the group stored in the global variable
<code>IMAGE_GROUP</code>. <font face="Gill Sans,Helvetica,Arial">GAP</font> finds three epimorphisms. Display the three
kernels by selecting <var>display</var> in the epimorphisms window.
<p>
Note that the new vertex <var>9</var> will be drawn on the line between
vertices <var>2</var> and <var>3</var> 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.
<p>
Now select vertices <var>8</var> and <var>11</var> and calculate the intersection of the 
two subgroups of indices <var>137438953472</var> and <var>6</var> respectively. <font face="Gill Sans,Helvetica,Arial">GAP</font>
can calculate this intersection by calculating the subdirect product
of the image groups of the epimorphisms (the index of the subgroup
belonging to vertex <var>12</var> in <var>G</var> is <var>412316860416</var> which is three times
of the index of the subgroup belonging to vertex <var>11</var>). Note that this 
subdirect product can only be calculated because the two image groups
are polycyclic groups. This is the reason why we needed <var>S<sub>3</sub></var> as
polycyclic group earlier.
<p>
This  is now  the end of   our partial investigation  of the (partial)
subgroup lattice  of <var>G</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
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