<html><head><title>[xgap] 4.3 A Partial Subgroup Lattice of the Cavicchioli Group</title></head>
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<h1>4.3 A Partial Subgroup Lattice of the Cavicchioli Group</h1><p>
<p>
This section investigates the following finitely presented group <var>C<sub>2</sub></var>,
which was first investigated by Alberto Cavicchioli in <a href="biblio.htm#Cav86"><cite>Cav86</cite></a>:
<p><var>
langlea, b ;;; aba<sup>-2</sup>ba=b, (b<sup>-1</sup>a<sup>3</sup>b<sup>-1</sup>a<sup>-3</sup>)<sup>2</sup>=a<sup>-1</sup>rangle.
<p></var>
<p>
In this example we will show a way to prove a finitely presented group
to be infinite, and to find some big nonabelian factor groups of it.
<p>
The following <font face="Gill Sans,Helvetica,Arial">GAP</font> commands define <var>C<sub>2</sub></var>.
<p>
<pre>
gap> f := FreeGroup( "a", "b" ); a := f.1;; b := f.2;;
<free group on the generators [ a, b ]>
gap> c2 := f / [ a*b*a^-2*b*a/b, (b^-1*a^3*b^-1*a^-3)^2*a ];
<fp group on the generators [ a, b ]>
gap> SetName(c2,"c2");
</pre>
<p>
We again assume that you are familiar with the general ideas, mouse actions
and menus, which were discussed in <a href="C004S001.htm">The Subgroup Lattice of the Dihedral Group of Order 8</a> and <a href="C004S002.htm">A Partial Subgroup Lattice of the Symmetric Group on 6 Points</a>.
<p>
In order to build a partial lattice of a finitely presented group, you
again use the function <code>GraphicSubgroupLattice</code>. But if the first argument
to <code>GraphicSubgroupLattice</code> is a finitely presented group the available menus
are different from the example in the previous section. After you have
entered
<p>
<pre>
gap> s := GraphicSubgroupLattice(c2);
<graphic subgroup lattice "GraphicSubgroupLattice of c2">
</pre>
<p>
XGAP will open a window containing a new graphic sheet. Compared to the
interactive lattice of a permutation group as described in the previous
section, there are the following differences:
<p>
-- There is only one vertex instead of two. This vertex labeled <var>G</var> is the
whole group <var>C<sub>2</sub></var>. There is no vertex for the trivial subgroup (yet).
<p>
-- If you pull down the <code>Subgroups</code> menu, you will see that this menu is
now very different. It gives you access to various algorithms for
finitely presented groups but most of the entries from the last two
examples are missing because most of the <font face="Gill Sans,Helvetica,Arial">GAP</font> functions behind these
entries are not applicable to (infinite) finitely presented groups.
<p>
This example will show you how to prove that <var>C<sub>2</sub></var> is infinite. First
look at the abelian invariants in order to see what the commutator factor
group is. In order to compute the abelian invariants pop up the
``Information'' menu. This is done in exactly the same manner as in the
previous section. Place the pointer inside vertex <var>G</var>, press the <strong>right</strong>
mouse button and release it immediately. This ``Information'' menu is
described in detail in <a href="C005S013.htm">GraphicSubgroupLattice for FpGroups, Information Menu</a>.
<p>
<pre>
Index 1
IsNormal true
IsFpGroup unknown
Abelian Invariants unknown
Coset Table unknown
IsomorphismFpGroup unknown
Factor Group unknown
</pre>
<p>
This tells you what XGAP already knows about the group associated with
vertex <var>G</var>. In order to compute the abelian invariants click onto this
line. After a while this entry will change to
<p>
<pre>
Abelian Invariants perfect
</pre>
<p>
telling you that <var>C<sub>2</sub></var> is perfect. So none of the <code>Subgroups</code> menu
entries <code>Abelian Prime Quotient</code>, <code>All Overgroups</code>, <code>Conjugacy Class</code>,
<code>Cores</code>, <code>Derived Subgroups</code>, <code>Intersection</code>, <code>Intersections</code>,
<code>Normalizers</code> or <code>Prime Quotient</code> will compute any new subgroups.
<p>
In order to avoid accidents the menu entries <code>Abelian Prime Quotient</code>,
<code>All Overgroups</code>, <code>Epimorphisms (GQuotients)</code>, <code>Conjugacy Class</code>,
<code>Low Index Subgroups</code>, and <code>Prime Quotient</code> from the <code>Subgroups</code> menu are
only selectable if exactly one vertex is selected because the
functions behind these entries are in general quite time and space
consuming.
<p>
Close the ``Information'' window and select <code>Low Index Subgroups</code> from the
<code>Subgroups</code> menu. A small dialog box will pop up asking for a limit on
the index. Type in <var>12</var> and press <var>return</var> or click on <code>OK</code>. In general
it is hard to say what kind of index limit will still work, for some
groups even <var>5</var> might be too much while for others <var>20</var> works fine, see
also <a href="../../../doc/htm/ref/C045S009.htm#SSEC1">LowIndexSubgroupsFpGroup</a>.
<p>
<font face="Gill Sans,Helvetica,Arial">GAP</font> computes <var>10</var> subgroups of index <var>11</var> and <var>8</var> subgroups of index
<var>12</var>. If you now start to check the abelian invariants of the index <var>12</var>
subgroups you will find out that all subgroups represented by vertices
<var>3</var> to <var>10</var> have a finite commutator factor group except the subgroup
belonging to vertex <var>4</var> which has an infinite abelian quotient.
Therefore the group <var>C<sub>2</sub></var> itself is infinite.
<p>
Now we want to investigate <var>C<sub>2</sub></var> a little further using <font face="Gill Sans,Helvetica,Arial">GAP</font>. Select
vertices <var>3</var>, <var>4</var>, and <var>5</var> and switch to the <font face="Gill Sans,Helvetica,Arial">GAP</font> window. Use
<code>SelectedGroups</code> to get the subgroups associated with these vertices.
<p>
<pre>
gap> u := SelectedGroups( s );
[ Group([ a, b*a^2*b^-2, b*a*b^2*a^-1*b^-1*a^-1*b^-1, b^4*a^-2*b^-2,
b^2*a^3*b^-1*a^-1*b^-2 ]),
Group([ a, b^2*a*b^-1*a^-1*b^-1, b^3*a^-1*b^-1, b*a*b*a^3*b^-1 ]),
Group([ a, b^2*a*b^-1*a^-1*b^-1, b*a^3*b^-2, b^4*a^-1*b^-3,
b*a*b^3*a^-1*b^-1 ]) ]
</pre>
<p>
<code>FactorCosetOperation</code> computes for each of these subgroups <var>u<sub>i</sub></var> the
operation of <var>C<sub>2</sub></var> on its cosets. It returns the result as a homomorphism
of <var>C<sub>2</sub></var> onto a permutation group. The operation on <var>u<sub>i</sub></var> is therefore a
permutation representation of the factor group <p><var>C<sub>2</sub> / Core(u<sub>i</sub>).<p></var> Using
<code>DisplayCompositionSeries</code> we can identify these factor groups.
<p>
<pre>
gap> p := List( u, x -> FactorCosetOperation( c2, x ) );;
gap> l := List( p, Image );;
gap> for x in l do DisplayCompositionSeries(x); Print("\n"); od;
G (2 gens, size 95040)
| M(12)
1 (0 gens, size 1)
G (2 gens, size 660)
| A(1,11) = L(2,11) ~ B(1,11) = O(3,11) ~ C(1,11) = S(2,11) ~ 2A(1,11) = U(2,11)
1 (0 gens, size 1)
G (2 gens, size 239500800)
| A(12)
1 (0 gens, size 1)
</pre>
<p>
(This display can look a little different according to the <font face="Gill Sans,Helvetica,Arial">GAP</font> version you
use.)
<p>
So <var>C<sub>2</sub></var> contains the Mathieu group <var>M<sub>12</sub></var>, the alternating group on
<var>12</var> symbols and <var>PSL(2,11)</var> as factor groups. Therefore it would
have been possible to find vertex <var>4</var> using
<code>Epimorphisms (GQuotients)</code> instead of <code>Low Index Subgroups</code>.
Close the graphic
sheet by selecting the menu entry <code>close graphic sheet</code> from the <code>Sheet</code>
menu and start with a fresh one.
<p>
<pre>
gap> s := GraphicSubgroupLattice(c2);
<graphic subgroup lattice "GraphicSubgroupLattice of c2">
</pre>
<p>
Select <code>Epimorphisms (GQuotients)</code> from the <code>Subgroups</code> menu. This
pops up a menu similar to the ``Information'' menu (see
<a href="C005S012.htm">GraphicSubgroupLattice for FpGroups, Subgroups Menu</a>).
<p>
<pre>
Sym(n)
Alt(n)
PSL(d,q)
Library
User Defined
</pre>
<p>
Select <var>PSL(d,q)</var>, which pops up a dialog box asking for a dimension.
Enter <code>2</code> and click on <var>OK</var>. Then a second dialog box pops up asking
for a field size. Enter <code>11</code> and click on <var>OK</var>. After a short time
of computation the display in the <code>Epimorphisms (GQuotients)</code> menu
changes and shows
<p>
<pre>
PSL(2,11) 1 found
</pre>
<p>
telling you, that <font face="Gill Sans,Helvetica,Arial">GAP</font> has found <var>1</var> epimorphism (up to inner
automorphisms of <var>PSL(2,11)</var>) from <var>C<sub>2</sub></var> onto <var>PSL(2,11)</var>. Click on
<var>display point stabilizer</var> to create a new vertex representing a subgroup
<var>u</var> such that the factor group of <var>C<sub>2</sub> / Core(u)</var> is isomorphic to
<var>PSL(2,11)</var>. You could have clicked on <var>display</var> to create a new vertex
representing the kernel of the epimorphism.
<p>
This is now the end of our partial investigation of the (partial)
subgroup lattice of <var>C<sub>2</sub></var>, you have seen that <var>C<sub>2</sub></var> is infinite and
contains <var>M<sub>12</sub></var>, <var>Alt(12)</var>, and <var>PSL(2,11)</var> as factor groups. 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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