<html><head><title>[XGAP] 5 Subgroup Lattices - Systematic Description</title></head>
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<h1>5 Subgroup Lattices - Systematic Description</h1><p>
<P>
<H3>Sections</H3>
<oL>
<li> <A HREF="CHAP005.htm#SECT001">GraphicSubgroupLattice</a>
<li> <A HREF="CHAP005.htm#SECT002">GraphicSubgroupLattice, Protocol of Group Theoretic Constructions</a>
<li> <A HREF="CHAP005.htm#SECT003">GraphicSubgroupLattice, Labelling of Levels</a>
<li> <A HREF="CHAP005.htm#SECT004">GraphicSubgroupLattice, Moving Vertices</a>
<li> <A HREF="CHAP005.htm#SECT005">GraphicSubgroupLattice, Selecting Vertices</a>
<li> <A HREF="CHAP005.htm#SECT006">GraphicSubgroupLattice, Inserting Vertices</a>
<li> <A HREF="CHAP005.htm#SECT007">GraphicSubgroupLattice, Sheet Menu</a>
<li> <A HREF="CHAP005.htm#SECT008">GraphicSubgroupLattice, Poset Menu</a>
<li> <A HREF="CHAP005.htm#SECT009">GraphicSubgroupLattice, Subgroups Menu</a>
<li> <A HREF="CHAP005.htm#SECT010">GraphicSubgroupLattice, Information Menu</a>
<li> <A HREF="CHAP005.htm#SECT011">Vertex Shapes</a>
<li> <A HREF="CHAP005.htm#SECT012">GraphicSubgroupLattice for FpGroups, Subgroups Menu</a>
<li> <A HREF="CHAP005.htm#SECT013">GraphicSubgroupLattice for FpGroups, Information Menu</a>
</ol><p>
<p>
In this chapter we give details about the various options and menus
available in a systematic way.
<p>
<p>
<h2><a name="SECT001">5.1 GraphicSubgroupLattice</a></h2>
<p><p>
<a name = "SSEC001.1"></a>
<li><code>GraphicSubgroupLattice( </code><var>g</var><code> )</code>
<p>
<code>GraphicSubgroupLattice</code> creates a new graphic sheet for the Hasse
diagram of the subgroup lattice of <var>g</var>. The next sections describe
how to select and move vertices and the following sections describe the
available menus.
<p>
<pre>
gap> GraphicSubgroupLattice( DihedralGroup(8) );
<graphic subgroup lattice "GraphicSubgroupLattice">
</pre>
<p>
<a name = "SSEC001.2"></a>
<li><code>GraphicSubgroupLattice( </code><var>g</var><code>, </code><var>width</var><code>, </code><var>height</var><code> )</code>
<p>
In this form <code>GraphicSubgroupLattice</code> creates a graphic sheet of initial
dimensions <var>width</var> times <var>height</var>. However it is still possible to change
these dimensions later using either the menu (described in
<a href="CHAP005.htm#SECT008">GraphicSubgroupLattice, Poset Menu</a>) or the operation <code>Resize</code> for graphic
sheets (see <a href="CHAP006.htm#SSEC001.14">Resize</a>).
<p>
This function is the same for all types of groups. It only behaves differently
according to some properties of the group. For example finitely presented
groups are treated differently because other algorithms apply in this case
and some of the standard ones are not feasible to be carried out.
<p>
In contrast to the <font face="Gill Sans,Helvetica,Arial">GAP</font>3 version of XGAP all graphic subgroup lattices
are interactive so you can always remove vertices. For a finite group you
can still get the full lattice by choosing <code>All Subgroups</code> for the whole
group.
<p>
<p>
<h2><a name="SECT002">5.2 GraphicSubgroupLattice, Protocol of Group Theoretic Constructions</a></h2>
<p><p>
labelloggingfacility
XGAP offers the possibility to write a protocol of the group
theoretic constructions you perform via mouse clicks. This is
convenient because otherwise you have no record of what you did. The
normal <font face="Gill Sans,Helvetica,Arial">GAP</font> command script cannot supply this because the menu
entries you select do not produce useful output there.
<p>
You start the protocol facility by selecting <code>Start logging</code> from the
<code>Subgroups</code> menu. You are prompted for the file name of the protocol with
a file selector box. After selecting <code>OK</code> in this box, all subsequent
actions are logged in the protocol file. You can switch this feature
off by selecting <code>Stop logging</code> from the <code>Subgroups</code> menu.
<p>
Note that when you select <code>SelectedGroups to GAP</code> from the <code>Subgroups</code>
menu, the usual <font face="Gill Sans,Helvetica,Arial">GAP</font> logging is also directed to the XGAP log
file. This is stopped when you select <code>InsertVertices from GAP</code> from
the <code>Subgroups</code> menu. The idea of this is that you also see the
<font face="Gill Sans,Helvetica,Arial">GAP</font> commands you issue within an XGAP session in the XGAP
protocol file, if you temporarily work with the keyboard instead of
the mouse. See <a href="CHAP005.htm#SSEC009.21">Start logging</a> and <a href="CHAP005.htm#SSEC009.22">Stop logging</a> for a description
of the menu entries.
<p>
<p>
<h2><a name="SECT003">5.3 GraphicSubgroupLattice, Labelling of Levels</a></h2>
<p><p>
labellevelsintro
We intend to represent subgroups of the same index at the same height
of the graphic lattice. For this purpose we introduce ``levels'' as
horizontal slices of a graphic subgroup lattice.
<p>
All vertices for subgroups of a certain finite index are
placed in exactly one common level. But what about infinite indices?
<p>
Every level has a ``level parameter''.
There are three types of levels: ``finite index'', ``finite size'', and
``infinity''. ``finite index'' means, that the index of the subgroups
in this level is a certain, finite natural number, the level parameter
is exactly this number. ``finite size'' means, that the size of the
subgroups in this level is finite <strong>and</strong> the index is either not known
(to <font face="Gill Sans,Helvetica,Arial">GAP</font>) or <code>infinity</code>. The level parameter is the size but with a
negative sign. If the index is <code>infinity</code> and the size is
not known, the third type applies: ``infinity''. In each ``infinity''
level only one vertex can exist.
<p>
This means that ``finite index'' takes precedence over ``finite size''
and ``infinity'', and ``finite size'' takes precedence over
``infinity'' if XGAP is in doubt which level to choose for a new
vertex.
<p>
If the group in question is a space group provided by the CRYST
package, levels of type ``infinity'' are also labelled by the Hirsch
length of the subgroup, which is the number of infinite cyclic factors
in any given subnormal series. Note that this is only implemented for
space groups as of this writing, although the Hirsch length is defined for
a wider range of groups.
<p>
For every graphic subgroup lattice the levels are partially ordered by
the following rules:
<p>
<ul>
<li> A ``finite index'' level is greater than an ``infinity'' level.
<p>
<li> An ``infinity'' is greater than a ``finite size'' level.
<p>
<li> The ``finite index'' levels are totally ordered by descending indices.
<p>
<li> The ``finite size'' levels are totally ordered by ascending sizes.
<p>
<li> For space groups the ``infinity'' levels are partially ordered by
the Hirsch lengths of the subgroups.
</ul>
<p>
Note that in general different ``infinity'' levels are not comparable in this
partial order!
<p>
On the screen, the levels are of course always totally ordered with respect
to their height. XGAP ensures that this total ordering is always
compatible with the abovementioned partial ordering. This means, that the
user can permute ``infinity'' levels, as long as this does not violate the
partial order by the Hirsch lengths and the
principle, that a vertex ``containing'' another one is drawn higher on
the screen.
<p>
Note that the level a vertex belongs to will be changed, when new
information about the subgroup is available. This happens if the user
chooses any entry in the information menu and new information is
available thereafter.
<p>
<p>
<h2><a name="SECT004">5.4 GraphicSubgroupLattice, Moving Vertices</a></h2>
<p><p>
In order to move a vertex, place the pointer inside the vertex using the
mouse, and press the <strong>left</strong> mouse button. Hold the mouse button pressed,
and start moving the pointer by moving the mouse. The vertex will now
follow the pointer, but it is not possible to move the vertex out of
the boundaries of its level.
or lower than a vertex of a group of smaller
size.
As soon as you release the left mouse
button the vertex will stop following the pointer and, if the vertex was
a member of a conjugacy class, the remaining elements of the class are
moved.
<p>
If you hold down the <var>SHIFT</var> key before moving a vertex as described
above then only the selected vertex is moved but not all members of the
same class. You can realign the vertices of a class by selecting one of
them and choose <code>Rearrange Classes</code> in the <code>Poset</code> menu
(see <a href="CHAP005.htm#SSEC008.14">Rearrange Classes</a>).
<p>
<p>
<h2><a name="SECT005">5.5 GraphicSubgroupLattice, Selecting Vertices</a></h2>
<p><p>
Selected vertices are represented by a slightly thicker circle and, if
your screen supports color, are colored red. There are five different
ways to select or deselect a vertex or a bunch of vertices.
<p>
Place the pointer inside a vertex and press the <strong>left</strong> mouse button.
Release the button immediately without moving the mouse. This will
deselect all other vertices and select this vertex. If the vertex was
already the only selected vertex it is deselected.
<p>
Place the pointer inside a vertex, hold down the <var>SHIFT</var> key, and press
the <strong>left</strong> mouse button. Release the button immediately without moving
the mouse, release the <var>SHIFT</var> key afterwards. If the vertex was
deselected this action will select it in addition to any other selected
vertices. If the vertex was already selected it will be deselected.
<p>
Place the pointer outside any vertex and press the <strong>left</strong> mouse button.
Keep the button pressed and start moving the pointer. This will create a
rubber band rectangle. One corner at the position where you pressed the
mouse button, the opposite corner following the pointer. As soon as you
release the left mouse button, all vertices inside the rectangle are
selected, all vertices outside the rectangle are deselected.
<p>
Place the pointer outside any vertex, hold down the <var>SHIFT</var> key and press
the <strong>left</strong> mouse button. Again you see a rubber band, however, now as
soon as you release the mouse button, all vertices inside the rectangle
are selected in addition to any other selected vertices.
<p>
To select vertices from the <font face="Gill Sans,Helvetica,Arial">GAP</font> command window call the function
<code>SelectGroups</code> with a subgroup or a list of subgroups to select.
<p>
<p>
<h2><a name="SECT006">5.6 GraphicSubgroupLattice, Inserting Vertices</a></h2>
<p><p>
To insert new vertices into a graphic sheet one has to use the
following operation:
<p>
<a name = "SSEC006.1"></a>
<li><code>InsertVertex( </code><var>sheet</var><code>, </code><var>grp</var><code>, </code><var>conj</var><code>, </code><var>hint</var><code> )</code>
<li><code>InsertVertex( </code><var>sheet</var><code>, </code><var>grp</var><code> )</code>
<p>
Inserts the group <var>grp</var> as a new vertex into the sheet. If the lattice
is <strong>not</strong> the subgroup lattice of a finitely presented group, it
checks, if the group is already in the lattice or if there is already
a conjugate subgroup. Further, the new vertex is sorted into the
poset. So <code>InsertVertex</code> checks for all vertices on higher levels,
whether the new vertex is contained, and for all vertices on lower
levels, whether they are contained in the new vertex. It then tries to
add edges to the appropriate vertices. If the lattice is the lattice
of a finitely presented group, nothing is done with respect to the
connections of any vertex. <code>InsertVertex</code> returns a list with the new
vertex as first entry and a flag as second, which says, whether this
vertex was inserted right now or has already been there.
<p>
<var>hint</var> is a list of <i>x</i> coordinates which should give some hint for
the choice of the new <i>x</i> coordinate. It can for example be the <i>x</i>
coordinates of those groups which were parameter for the operation
which calculated the group. <var>hints</var> can be empty but must always be a
list!
<p>
If <var>conj</var> is a vertex we put the new vertex into the class of this
vertex. Otherwise <var>conj</var> should either be <code>false</code> or <code>fail</code>.
<p>
You have access to the simpler form of the above functions via the
<code>Subgroups</code> menu and the mouse. If you select
<code>InsertVertices from GAP</code>
in the <code>Subgroups</code> menu, the function <code>InsertVertex</code> is called
for the group in the (automatic) variable <code>last</code>, or, if <code>last</code>
contains a list of subgroups, for all of those subgroups. This is an
easy way to insert results of calculations with <font face="Gill Sans,Helvetica,Arial">GAP</font> into the
lattice. If <code>last</code> is neither a subgroup nor a list of subgroups,
nothing happens.
<p>
<p>
<h2><a name="SECT007">5.7 GraphicSubgroupLattice, Sheet Menu</a></h2>
<p><p>
The <code>Sheet</code> menu will be pulled down if you place the pointer inside the
<code>Sheet</code> button and press the left mouse button. Keep the button down and
choose an entry by moving the pointer on top of this entry. Release the
mouse button to select an entry.
<p>
<a name = "SSEC007.1"></a>
<li><code>save as postscript</code>
<p>
Selecting this entry pops up a file selector. If you enter a file name
and click on <var>OK</var>, this will save a description of the graphic sheet and
graphic objects on the sheet as encapsulated postscript output. This
output should be imported easily into other documents.
<p>
<a name = "SSEC007.2"></a>
<li><code>close graphic sheet</code>
<p>
This entry will close the current graphic sheet and the window containing
the sheet and all ``Information'' menus related to this sheet. On some
window system or using some window manager the window itself has also a
close button or menu entry. However, using the window close method will
not close associated ``Information'' menus.
<p>
<p>
<h2><a name="SECT008">5.8 GraphicSubgroupLattice, Poset Menu</a></h2>
<p><p>
The <code>Poset</code> menu will be pulled down if you place the pointer inside the
<code>Poset</code> button and press the left mouse button. Keep the button down
and choose an entry by moving the pointer on top of this entry. Release
the mouse button to select an entry.
<p>
This menu is a generic menu for any graphic poset and is not specific to
subgroup lattices. So you find here options for the handling of general
posets.
<p>
<a name = "SSEC008.1"></a>
<li><code>Redraw</code>
<p>
The whole lattice will be redrawn. Use this option if some manipulation
has disturbed the window.
<p>
<a name = "SSEC008.2"></a>
<li><code>Show Levels</code>
<p>
If this menu entry is activated (after clicking it will have a small
check sign at its right, clicking again deactivates it), there will be a
little blue box under each level at the left edge of the graphic
sheet. You can use these little boxes to change the height of a level by
moving it up or down like a vertex. In cases where the order of the
levels is not given by some ``external source'' like the index of the
subgroups you can move levels by moving the corresponding blue box with
pressed <var>SHIFT</var> button.
<p>
<a name = "SSEC008.3"></a>
<li><code>Show Levelparameters</code>
<p>
This menu entry controls the display of the level parameters at the right
edge of the graphic sheet. Normally these are displayed but you can
switch it off with this menu entry.
<p>
<a name = "SSEC008.4"></a>
<li><code>Delete Vertices</code>
<p>
Selecting the entry will delete all selected vertices. All edges from or
to one of these are also deleted. However, inclusion information is
preserved. This means that new edges are created from all vertices which
were maximal in the deleted one to all vertices in which the deleted
vertex was maximal. So you get the Hasse diagram of the poset which is
the restriction of the former one to the not selected vertices.
<p>
<a name = "SSEC008.5"></a>
<li><code>Delete Edge</code>
<p>
This menu entry is normally not selectable because it would destroy the
Hasse diagram.
<p>
<a name = "SSEC008.6"></a>
<li><code>Merge Classes</code>
<p>
This menu entry merges all classes within each level that contain a
selected vertex. Afterwards <code>RearrangeClasses</code> (see below) is
performed. Use <code>Merge Classes</code> to unite classes within a level, but
use it with care: No further test is performed! If you unite classes
of subgroups that are not conjugate, other errors may follow, because
conjugacy tests are later on only performed by looking at the first
subgroup in a given class!
<p>
<a name = "SSEC008.7"></a>
<li><code>Magnify Lattice</code>
<p>
Selecting the entry will multiply the dimensions of the graphic sheet by
the square root of 2 and enlarge the lattice accordingly.
<p>
<a name = "SSEC008.8"></a>
<li><code>Shrink Lattice</code>
<p>
Selecting the entry will divide the dimensions of the graphic sheet by
the square root of 2 and shrink the lattice accordingly.
<p>
<a name = "SSEC008.9"></a>
<li><code>Resize Lattice</code>
<p>
Selecting this entry will pop up a dialog box asking for an x and y factor
separated by a comma. You can enter integers or quotients. If you enter
only one number this is used for x and y. The graphic sheet is then
enlarged or shrinked and the lattice is resized accordingly.
<p>
<a name = "SSEC008.10"></a>
<li><code>Resize Sheet</code>
<p>
This entry is similar to <code>Resize Lattice</code> except that only the graphic
sheet is changed, the lattice remains unchanged. The numbers you enter must
be integers and mean pixel numbers.
<p>
<a name = "SSEC008.11"></a>
<li><code>Change Labels</code>
<p>
Selecting this entry will pop up a dialog box for each selected vertex
asking for a new label. Clicking on <var>CANCEL</var> will cancel the relabelling
of the remaining vertices but will not reset the already changed labels.
<p>
<a name = "SSEC008.12"></a>
<li><code>Average Y Positions</code>
<p>
Selecting this entry will average the y coordinates of all vertices
belonging to the same level. For graphic subgroup lattices this means all
subgroups with the same index in the whole group.
<p>
<a name = "SSEC008.13"></a>
<li><code>Average X Positions</code>
<p>
Selecting this entry will average the x coordinates of two or more
selected vertices. This will only work if the corresponding subgroups do
not have the same size and is used to align certain vertices
vertically.
<p>
<a name = "SSEC008.14"></a>
<li><code>Rearrange Classes</code>
<p>
Selecting this entry will clean up all classes which contain a selected
vertex. You need this option if you have moved a vertex without its class
(holding down the <var>SHIFT</var> key). The vertices in a class are arranged one
next to the other horizontally without changing the order of the x
coordinates. So you can permute the vertices within a class carelessly and
then again get a nice picture with the new order by selecting this menu
entry.
<p>
<a name = "SSEC008.15"></a>
<li><code>Use Black&White</code>
<p>
Switches to black and white in case of a color screen or back to colors.
<p>
<p>
<h2><a name="SECT009">5.9 GraphicSubgroupLattice, Subgroups Menu</a></h2>
<p><p>
The <code>Subgroups</code> menu will be pulled down if you place the pointer inside
the <code>Subgroups</code> button and press the left mouse button. Keep the button
down and choose an entry by moving the pointer on top of this entry.
Release the mouse button to select an entry.
<p>
Note that you can also get the <code>Subgroups</code> menu as a popup menu by
clicking with the right mouse button into the graphic sheet of the subgroup
lattice, but <strong>not</strong> on a vertex.
<p>
The result of a computation from any of the following entries is
colored green, if your screen supports color. There will also be a short
information message in the <font face="Gill Sans,Helvetica,Arial">GAP</font> window about the result.
<p>
In the following descriptions we use ``vertices'' as abbreviation for
``subgroups associated with vertices''.
<p>
The following descriptions do not apply to the case of finitely presented
groups. See <a href="CHAP005.htm#SECT012">GraphicSubgroupLattice for FpGroups, Subgroups Menu</a> for this
case.
<p>
<a name = "SSEC009.1"></a>
<li><code>All Subgroups</code>
<p>
For each selected vertex <code>All Subgroups</code> computes and displays all its
subgroups. Requires at least one selected vertex. Use with care! This can
cause huge computations! See also <a href="../../../doc/htm/ref/CHAP037.htm#SSEC019.1">LatticeSubgroups</a> in the <font face="Gill Sans,Helvetica,Arial">GAP</font>
reference manual.
<p>
<a name = "SSEC009.2"></a>
<li><code>Centralizers</code>
<p>
For each selected vertex <code>Centralizers</code> computes and displays its
centralizer with respect to the whole group. Requires at least one
selected vertex. See also <a href="../../../doc/htm/ref/CHAP033.htm#SSEC004.4">Centralizer</a> in the <font face="Gill Sans,Helvetica,Arial">GAP</font>
reference manual.
<p>
<a name = "SSEC009.3"></a>
<li><code>Centres</code>
<p>
For each selected vertex <code>Centres</code> computes and displays its centre.
Requires at least one selected vertex. See also <a href="../../../doc/htm/ref/CHAP033.htm#SSEC004.5">Centre</a> in the <font face="Gill Sans,Helvetica,Arial">GAP</font>
reference manual.
<p>
<a name = "SSEC009.4"></a>
<li><code>Closure</code>
<p>
computes and displays the common closure of the selected vertices.
Requires at least one selected vertex. See also <a href="../../../doc/htm/ref/CHAP037.htm#SSEC004.1">ClosureGroup</a> in the
<font face="Gill Sans,Helvetica,Arial">GAP</font> reference manual.
<p>
<a name = "SSEC009.5"></a>
<li><code>Closures</code>
<p>
computes and displays the closure of each pair of selected vertices.
Requires at least two selected vertices. See also <a href="../../../doc/htm/ref/CHAP037.htm#SSEC004.1">ClosureGroup</a> in the
<font face="Gill Sans,Helvetica,Arial">GAP</font> reference manual.
<p>
<a name = "SSEC009.6"></a>
<li><code>Commutator Subgroups</code>
<p>
computes and displays the commutator subgroup of each pair of selected
vertices. Requires at least two selected vertices. See also
<a href="../../../doc/htm/ref/CHAP037.htm#SSEC011.2">CommutatorSubgroup</a> in the <font face="Gill Sans,Helvetica,Arial">GAP</font>
reference manual.
<p>
<a name = "SSEC009.7"></a>
<li><code>Conjugate Subgroups</code>
<p>
computes and displays the conjugacy class (with respect to the whole
group) of each selected vertex. Requires at least one selected vertex.
See also <a href="../../../doc/htm/ref/CHAP037.htm#SSEC009.1">ConjugacyClass</a> in the <font face="Gill Sans,Helvetica,Arial">GAP</font>
reference manual.
<p>
<a name = "SSEC009.8"></a>
<li><code>Cores</code>
<p>
For each selected vertex <code>Cores</code> computes and displays its core with
respect to the whole group. Requires at least one selected vertex. See
also <a href="../../../doc/htm/ref/CHAP037.htm#SSEC010.2">Core</a> in the <font face="Gill Sans,Helvetica,Arial">GAP</font>
reference manual.
<p>
<a name = "SSEC009.9"></a>
<li><code>Derived Series</code>
<p>
For each selected vertex <code>Derived Series</code> computes and displays its
derived series. Requires at least one selected vertex. See also
<a href="../../../doc/htm/ref/CHAP037.htm#SSEC016.7">DerivedSeriesOfGroup</a> in the <font face="Gill Sans,Helvetica,Arial">GAP</font>
reference manual.
<p>
<a name = "SSEC009.10"></a>
<li><code>Derived Subgroups</code>
<p>
For each selected vertex <code>Derived Subgroups</code> computes and displays its
derived subgroup. Requires at least one selected vertex. See also
<a href="../../../doc/htm/ref/CHAP037.htm#SSEC011.3">DerivedSubgroup</a> in the <font face="Gill Sans,Helvetica,Arial">GAP</font>
reference manual.
<p>
<a name = "SSEC009.11"></a>
<li><code>Fitting Subgroups</code>
<p>
For each selected vertex <code>Fitting Subgroups</code> computes and displays its
Fitting subgroup. Requires at least one selected vertex. See also
<a href="../../../doc/htm/ref/CHAP037.htm#SSEC011.5">FittingSubgroup</a> in the <font face="Gill Sans,Helvetica,Arial">GAP</font>
reference manual.
<p>
<a name = "SSEC009.12"></a>
<li><code>Intermediate Subgroups</code>
<p>
computes and displays all intermediate subgroups between two selected
groups. Requires exactly two selected vertices. See also
<a href="../../../doc/htm/ref/CHAP037.htm#SSEC016.18">IntermediateSubgroups</a> in the <font face="Gill Sans,Helvetica,Arial">GAP</font>
reference manual.
<p>
<a name = "SSEC009.13"></a>
<li><code>Intersection</code>
<p>
computes and displays the common intersection of the selected vertices.
Requires at least one selected vertex. See also <a href="../../../doc/htm/ref/CHAP028.htm#SSEC004.2">Intersection</a> in the
<font face="Gill Sans,Helvetica,Arial">GAP</font> reference manual.
<p>
<a name = "SSEC009.14"></a>
<li><code>Intersections</code>
<p>
For each pair of selected vertices <code>Intersections</code> computes and displays
the intersection of the two vertices. Requires at least two selected
vertices. See also <a href="../../../doc/htm/ref/CHAP028.htm#SSEC004.2">Intersection</a> in the <font face="Gill Sans,Helvetica,Arial">GAP</font>
reference manual.
<p>
<a name = "SSEC009.15"></a>
<li><code>Normalizers</code>
<p>
For each selected vertex <code>Normalizers</code> computes and displays its normalizer
with respect to the whole group. Requires at least one selected vertex.
See also <a href="../../../doc/htm/ref/CHAP037.htm#SSEC010.1">Normalizer</a> in the <font face="Gill Sans,Helvetica,Arial">GAP</font>
reference manual.
<p>
<a name = "SSEC009.16"></a>
<li><code>Normal Closures</code>
<p>
For each selected vertex <code>Normal Closure</code> computes and displays its normal
closure with respect to the whole group. Requires at least one selected
vertex. See also <a href="../../../doc/htm/ref/CHAP037.htm#SSEC010.4">NormalClosure</a> in the <font face="Gill Sans,Helvetica,Arial">GAP</font>
reference manual.
<p>
<a name = "SSEC009.17"></a>
<li><code>Normal Subgroups</code>
<p>
For each selected vertex <code>Normal Subgroups</code> computes and displays the
normal subgroups of the subgroup associated with this vertex. These new
subgroups are not necessarily normal in the whole group. Requires at
least one selected vertex. See also <a href="../../../doc/htm/ref/CHAP037.htm#SSEC018.7">NormalSubgroups</a> in the <font face="Gill Sans,Helvetica,Arial">GAP</font>
reference manual.
<p>
<a name = "SSEC009.18"></a>
<li><code>Sylow Subgroups</code>
<p>
pops up a dialog box asking for a prime. After entering a prime <i>p</i>
and pressing <var>return</var> or clicking <var>OK</var> it computes and displays a Sylow
<i>p</i>-subgroup for each selected vertex. Requires at least one selected
vertex. See also <a href="../../../doc/htm/ref/CHAP037.htm#SSEC012.1">SylowSubgroup</a> in the <font face="Gill Sans,Helvetica,Arial">GAP</font>
reference manual.
<p>
<a name = "SSEC009.19"></a>
<li><code>SelectedGroups to GAP</code>
<p>
If the user selects this menu entry, the subgroups belonging to the
selected vertices are put into a list which is stored into the variable
<code>last</code>. This is equivalent to the statement <code>SelectedGroups(sheet);;</code> if
<code>sheet</code> contains the graphic sheet object. If XGAP logging is on, then
the normal <font face="Gill Sans,Helvetica,Arial">GAP</font> logging via <code>LogTo</code> is also directed to the XGAP log
file.
<p>
<a name = "SSEC009.20"></a>
<li><code>InsertVertices from GAP</code>
<p>
If the user selects this menu entry, the value of the variable <code>last</code> is
used to insert new vertices into the graphic sheet. If <code>last</code> is equal to
one subgroup, it is inserted via <code>InsertVertex</code>. If <code>last</code> is a list of
subgroups, <code>InsertVertex</code> is called for all those subgroups. There is no
error issued if one of the entries of <code>last</code> is no subgroup. If XGAP
logging is on, then the normal <font face="Gill Sans,Helvetica,Arial">GAP</font> logging via <code>LogTo</code> is switched off!
The idea of this is to switch the logging temporarily from XGAP logging
to normal <font face="Gill Sans,Helvetica,Arial">GAP</font> logging between two clicks to <code>SelectedGroups to GAP</code>
and <code>InsertVertices from GAP</code> respectively.
<p>
<a name = "SSEC009.21"></a>
<li><code>Start Logging</code>
<p>
After clicking on this menu entry the user is prompted for a filename. From
this point on all commands issued via mouse clicks in the subgroup menu are
logged into that file, such that one can afterwards see ``what happened''
in the XGAP session. The information displayed is the same as in the
info displays in the <font face="Gill Sans,Helvetica,Arial">GAP</font> window.
<p>
<a name = "SSEC009.22"></a>
<li><code>Stop Logging</code>
<p>
A click onto this menu entry stops the XGAP logging.
<p>
These menu entries represent only a small selection of the functions
of <font face="Gill Sans,Helvetica,Arial">GAP</font> which the authors of XGAP considered most frequently
used. You can calculate other subgroups like for example prefrattini
subgroups from the <font face="Gill Sans,Helvetica,Arial">GAP</font> command window. See sections <a href="CHAP004.htm#SECT001">gapxgap</a> and
<a href="CHAP004.htm#SECT002">xgapgap</a> for examples how to transfer information from the graphical
lattice of XGAP to <font face="Gill Sans,Helvetica,Arial">GAP</font> (via <code>SelectedGroups</code>, see
<a href="CHAP005.htm#SECT005">GraphicSubgroupLattice, Selecting Vertices</a>) and vice versa (via
<code>SelectGroups</code>, see <a href="CHAP005.htm#SECT005">GraphicSubgroupLattice, Selecting Vertices</a>, and
<code>InsertVertex</code>, see <a href="CHAP005.htm#SECT006">GraphicSubgroupLattice, Inserting Vertices</a>).
<p>
<p>
<h2><a name="SECT010">5.10 GraphicSubgroupLattice, Information Menu</a></h2>
<p><p>
Note that this section does not deal with the case of a finitely presented
group. See <a href="CHAP005.htm#SECT013">GraphicSubgroupLattice for FpGroups, Information Menu</a> for this
case.
<p>
Placing the pointer inside a vertex (selected or not) and pressing the
<strong>right</strong> mouse button pops up the ``Information'' menu. Clicking on any of
the text lines will compute the corresponding property of the subgroup
<i>u</i> associated with this vertex. Clicking on <code>all</code> will compute all
properties, clicking on <code>close</code> will close the ``Information'' menu.
<p>
<a name = "SSEC010.1"></a>
<li><code>Size</code>
<p>
computes and displays the size of <i>u</i>. See also <a href="../../../doc/htm/ref/CHAP028.htm#SSEC003.6">Size</a> in the
<font face="Gill Sans,Helvetica,Arial">GAP</font> reference manual.
<p>
<a name = "SSEC010.2"></a>
<li><code>Index</code>
<p>
computes and displays the index of <i>u</i> in the whole group. See also
<a href="../../../doc/htm/ref/CHAP037.htm#SSEC003.3">IndexInWholeGroup</a> in the <font face="Gill Sans,Helvetica,Arial">GAP</font> reference manual.
<p>
<a name = "SSEC010.3"></a>
<li><code>IsAbelian</code>
<p>
<a name = "SSEC010.4"></a>
<li><code>IsCyclic</code>
<p>
<a name = "SSEC010.5"></a>
<li><code>IsNilpotent</code>
<p>
<a name = "SSEC010.6"></a>
<li><code>IsPerfect</code>
<p>
<a name = "SSEC010.7"></a>
<li><code>IsSimple</code>
<p>
<a name = "SSEC010.8"></a>
<li><code>IsSolvable</code>
<p>
computes and displays the corresponding property of <i>u</i>. See also
<a href="../../../doc/htm/ref/CHAP033.htm#SSEC004.9">IsAbelian</a>, <a href="../../../doc/htm/ref/CHAP037.htm#SSEC014.1">IsCyclic</a>, <a href="../../../doc/htm/ref/CHAP037.htm#SSEC014.3">IsNilpotentGroup</a>, <a href="../../../doc/htm/ref/CHAP037.htm#SSEC014.5">IsPerfectGroup</a>,
<a href="../../../doc/htm/ref/CHAP037.htm#SSEC014.10">IsSimpleGroup</a>, and <a href="../../../doc/htm/ref/CHAP037.htm#SSEC014.6">IsSolvableGroup</a> in the <font face="Gill Sans,Helvetica,Arial">GAP</font>
reference manual.
<p>
<a name = "SSEC010.9"></a>
<li><code>IsCentral</code>
<p>
<a name = "SSEC010.10"></a>
<li><code>IsNormal</code>
<p>
computes and displays the corresponding property of <i>u</i> with respect to
the whole group. See also <a href="../../../doc/htm/ref/CHAP033.htm#SSEC004.8">IsCentral</a> and <a href="../../../doc/htm/ref/CHAP037.htm#SSEC003.6">IsNormal</a> in the <font face="Gill Sans,Helvetica,Arial">GAP</font>
reference manual.
<p>
<a name = "SSEC010.11"></a>
<li><code>Isomorphism</code>
<p>
computes and displays the isomorphism type of <i>u</i>. This will only work
if the size of <i>u</i> is small. See <a href="badlink:ref:IdGroup">IdGroup</a> in the <font face="Gill Sans,Helvetica,Arial">GAP</font>
reference manual for details.
<p>
Note that the exact result of all these information displays is stored in
the global variable <code>LastResultOfInfoDisplay</code> after each operation. So you
can access this easily from the <font face="Gill Sans,Helvetica,Arial">GAP</font> command prompt. It is also returned
as <code>last</code> value.
<p>
<p>
<h2><a name="SECT011">5.11 Vertex Shapes</a></h2>
<p><p>
The following vertex shapes can appear in an interactive lattice:
<p>
<a name = "SSEC011.1"></a>
<li><code>circle</code>
<p>
Subgroup is not normal in whole group.
<p>
<a name = "SSEC011.2"></a>
<li><code>diamond</code>
<p>
Subgroup is normal in whole group.
<p>
<a name = "SSEC011.3"></a>
<li><code>rectangle</code>
<p>
Subgroup has an index that is too big for automatic calculation of
normality. So it is not yet known whether this group is normal.
<p>
Automatic calculation is controlled by the following variable:
<p>
<a name = "SSEC011.4"></a>
<li><code>GGLLimitForIsNormalCalc V</code>
<p>
Only for subgroups with index smaller than this number an automatic
<code>IsNormal</code> test is performed when the vertex is added to the sheet.
<p>
<p>
<h2><a name="SECT012">5.12 GraphicSubgroupLattice for FpGroups, Subgroups Menu</a></h2>
<p><p>
The <code>Subgroups</code> menu will be pulled down if you place the pointer inside
the <code>Subgroups</code> button and press the left mouse button. Keep the button
down and choose an entry by moving the pointer on top of this entry.
Release the mouse button to select an entry.
<p>
Note that you can also get the <code>Subgroups</code> menu as a popup menu by
clicking with the right mouse button into the graphic sheet of the subgroup
lattice, but <strong>not</strong> on a vertex.
<p>
The result of a computation from any of the following entries is
colored green, if your screen supports color. In most cases there will
also be short information message in the <font face="Gill Sans,Helvetica,Arial">GAP</font> window about the result.
<p>
Note that some of the menu entries make it necessary to compute
presentations of subgroups using a modified Todd-Coxeter algorithm.
This can be very time consuming and in some cases even impossible, if
the index is too high.
<p>
In the following descriptions, we use ``vertices'' as abbreviation for
``subgroups associated with vertices''.
<p>
<a name = "SSEC012.1"></a>
<li><code>Abelian Prime Quotient</code>
<p>
pops up a dialog box asking for a prime <i>p</i>. It then computes and
displays the largest elementary abelian <i>p</i> quotient of the selected
vertex. If no presentation for the subgroup associated to the vertex is
known a presentation is first computed using a modified Todd-Coxeter
algorithm. It then calls <code>PrimeQuotient</code> to compute the largest
elementary abelian quotient. <code>Abelian PrimeQuotient</code> requires exactly one
selected vertex.
<p>
<a name = "SSEC012.2"></a>
<li><code>All Overgroups</code>
<p>
computes and displays all overgroups of the selected vertex. It first
computes the permutation action of the whole group on the cosets of the
subgroup associated with the selected vertex and then searches for all
block systems. If the subgroup of the selected vertex is normal, then
everything is calculated within the (finite) factor group in a better
representation. <code>All Overgroups</code> requires exactly one selected vertex.
<p>
<a name = "SSEC012.3"></a>
<li><code>Closure</code>
<p>
computes and displays the common closure of the selected vertices.
Requires at least one selected vertex. See also <a href="../../../doc/htm/ref/CHAP037.htm#SSEC004.1">ClosureGroup</a> in the
<font face="Gill Sans,Helvetica,Arial">GAP</font> reference manual.
<p>
<a name = "SSEC012.4"></a>
<li><code>Compare Subgroups</code>
<p>
A non-empty set of vertices must be selected to choose this menu entry. All
subgroups belonging to these vertices are compared pairwise, and the inclusion
information is displayed in the lattice. It may happen that two or more
vertices are merged if <font face="Gill Sans,Helvetica,Arial">GAP</font> notices, that the subgroups are
equal.
<p>
<a name = "SSEC012.5"></a>
<li><code>Conjugacy Class</code>
<p>
computes and displays the conjugacy class of the selected vertex.
<code>Conjugacy Class</code> requires exactly one selected vertex.
<p>
<a name = "SSEC012.6"></a>
<li><code>Cores</code>
<p>
computes and displays the cores of the selected vertices. <code>Cores</code>
requires at least one selected vertex.
<p>
<a name = "SSEC012.7"></a>
<li><code>Derived Subgroups</code>
<p>
computes and displays the derived subgroups of the selected vertices.
If applied to a proper subgroup of the whole group it will only
display those derived subgroups whose index is finite. <code>Derived
Subgroups</code> requires at least one selected vertex.
<p>
<a name = "SSEC012.8"></a>
<li><code>Epimorphisms (GQuotients)</code>
<p>
pops up another menu. Requires exactly one selected vertex.
<p>
<pre>
Sym(n)
Alt(n)
PSL(d,q)
Library
User Defined
</pre>
<p>
Click on any of these entries to try to find a quotient isomorphic to the
symmetric group (<code>Sym(n)</code>), the alternating group (<code>Alt(n)</code>), the projective
special linear group (<code>PSL(d,q)</code>), a group in a library supplied with
XGAP (this will pop up a file selector), or a user defined group stored
in the variable <code>IMAGE_GROUP</code>. After supplying additional parameters, for
example, the degree of the symmetric group or the dimension and field of
<i>PSL</i> using dialog boxes, the corresponding entry will change, for example
to something like
<p>
<pre>
Sym(3) 3 found
</pre>
<p>
After one or more quotients were found click <var>display</var> to display them.
<p>
Note that in XGAP4 in fact the kernel of the epimorphism is marked whereas
in XGAP3 this was not the case, even though the XGAP3 manual stated
this.
<p>
In fact in XGAP3 a stabilizer of a permutation action on an orbit was
put into the lattice.
<p>
In case that the image of the epimorphism is a permutation group you can
get this functionality by clicking on <var>display point stabilizer</var> instead of
<var>display</var>.
<p>
<a name = "SSEC012.9"></a>
<li><code>Intermediate Subgroups</code>
<p>
computes and displays all intermediate subgroups between two selected
groups. Requires exactly two selected vertices. See also
<a href="../../../doc/htm/ref/CHAP037.htm#SSEC016.18">IntermediateSubgroups</a> in the <font face="Gill Sans,Helvetica,Arial">GAP</font> reference manual.
<p>
<a name = "SSEC012.10"></a>
<li><code>Intersection</code>
<p>
computes and displays the common intersection of the selected vertices.
Requires at least one selected vertex. See also <a href="../../../doc/htm/ref/CHAP028.htm#SSEC004.2">Intersection</a> in the
<font face="Gill Sans,Helvetica,Arial">GAP</font> reference manual.
<p>
<a name = "SSEC012.11"></a>
<li><code>Intersections</code>
<p>
computes and displays the pairwise intersections of the selected
vertices. <code>Intersections</code> requires at least two selected vertices.
<p>
<a name = "SSEC012.12"></a>
<li><code>Low Index Subgroups</code>
<p>
pops up a dialog box asking for index limit <i>k</i>. It will then do a low
index subgroup search for subgroups of index at most <i>k</i> of the selected
vertex using <code>LowIndexSubgroupsFpGroup</code>. If no presentation for the
subgroup associated to the vertex is known a presentation is first
computed using a modified Todd-Coxeter algorithm. <code>Low Index Subgroups</code>
requires exactly one selected vertex.
<p>
<a name = "SSEC012.13"></a>
<li><code>Normalizers</code>
<p>
computes and displays the normalizers of the selected vertices.
<code>Normalizers</code> requires at least one selected vertex.
<p>
<a name = "SSEC012.14"></a>
<li><code>Prime Quotient</code>
<p>
pops up a dialog box asking for a prime <i>p</i> and another dialog box asking
for a class <i>c</i>. It then computes and displays the largest <i>p</i>-quotient
of class <i>c</i> of the selected vertex. If no presentation for the subgroup
associated to the vertex is known a presentation is first computed using
a modified Todd-Coxeter algorithm. It then calls <code>PrimeQuotient</code>.
<code>Prime Quotient</code> requires exactly one selected vertex.
<p>
<a name = "SSEC012.15"></a>
<li><code>Test Conjugacy</code>
<p>
walks through all levels and tests for all pairs of classes, that contain a
selected vertex, whether the groups in the classes are conjugates. If so,
the classes are merged. After these calculations <code>Rearrange Classes</code> is
called. Note that conjugacy calculations can take lots of time for finitely
presented groups!
<p>
<a name = "SSEC012.16"></a>
<li><code>SelectedGroups to GAP</code>
<p>
If the user selects this menu entry, the subgroups belonging to the
selected vertices are put into a list which is stored into the variable
<code>last</code>. This is equivalent to the statement <code>SelectedGroups(sheet);;</code> if
<code>sheet</code> contains the graphic sheet object. If XGAP logging is on, then
the normal <font face="Gill Sans,Helvetica,Arial">GAP</font> logging via <code>LogTo</code> is also directed to the XGAP log
file.
<p>
<a name = "SSEC012.17"></a>
<li><code>InsertVertices from GAP</code>
<p>
If the user selects this menu entry, the value of the variable <code>last</code> is
used to insert new vertices into the graphic sheet. If <code>last</code> is equal to
one subgroup, it is inserted via <code>InsertVertex</code>. If <code>last</code> is a list of
subgroups, <code>InsertVertex</code> is called for all those subgroups. There is no
error issued if one of the entries of <code>last</code> is no subgroup. If XGAP
logging is on, then the normal <font face="Gill Sans,Helvetica,Arial">GAP</font> logging via <code>LogTo</code> is switched off!
The idea of this is to switch the logging temporarily from XGAP logging
to normal <font face="Gill Sans,Helvetica,Arial">GAP</font> logging between two clicks to ``SelectedGroups to GAP''
and ``InsertVertices from GAP'' respectively.
<p>
<a name = "SSEC012.18"></a>
<li><code>Start Logging</code>
<p>
After clicking on this menu entry the user is prompted for a filename. From
this point on all commands issued via mouse clicks in the subgroup menu are
logged into that file, such that one can afterwards see ``what happened''
in the XGAP session. The information displayed is the same as in the
info displays in the <font face="Gill Sans,Helvetica,Arial">GAP</font> window.
<p>
<a name = "SSEC012.19"></a>
<li><code>Stop Logging</code>
<p>
A click onto this menu entry stops the XGAP logging.
<p>
These menu entries represent only a small selection of the functions of
<font face="Gill Sans,Helvetica,Arial">GAP</font> which the authors of XGAP considered most frequently used. You
can calculate other subgroups from
the <font face="Gill Sans,Helvetica,Arial">GAP</font> command window. See sections <a href="CHAP004.htm#SECT001">gapxgap</a> and
<a href="CHAP004.htm#SECT002">xgapgap</a> for examples how to transfer information from the graphical
lattice of XGAP to <font face="Gill Sans,Helvetica,Arial">GAP</font> (via <code>SelectedGroups</code>, see
<a href="CHAP005.htm#SECT005">GraphicSubgroupLattice, Selecting Vertices</a>) and vice versa (via
<code>SelectGroups</code>, see <a href="CHAP005.htm#SECT005">GraphicSubgroupLattice, Selecting Vertices</a>, and
<code>InsertVertex</code>, see <a href="CHAP005.htm#SECT006">GraphicSubgroupLattice, Inserting Vertices</a>).
<p>
<p>
<h2><a name="SECT013">5.13 GraphicSubgroupLattice for FpGroups, Information Menu</a></h2>
<p><p>
Placing the pointer inside a vertex (selected or not) and pressing the
<strong>right</strong> mouse button pops up the ``Information'' menu. Clicking on any of
the text lines will compute the corresponding property of the subgroup
<i>u</i> associated with this vertex. Clicking on <code>close</code> will close the
``Information'' menu.
<p>
<a name = "SSEC013.1"></a>
<li><code>Index</code>
<p>
displays the index of <i>u</i> in the whole group.
<p>
<a name = "SSEC013.2"></a>
<li><code>IsNormal</code>
<p>
checks if <i>u</i> is normal in the whole group.
<p>
<a name = "SSEC013.3"></a>
<li><code>IsFpGroup</code>
<p>
checks if <i>u</i> is a finitely presented group. Note that a subgroup of a
finitely presented group that is defined by a coset table or as kernel
of an epimorphism is <strong>not</strong> automatically known to <font face="Gill Sans,Helvetica,Arial">GAP</font> as a finitely
presented group. This means, that certain algorithms can not be
applied. Use <code>IsomorphismFpGroup</code> (see <a href="../../../doc/htm/ref/CHAP045.htm#SSEC010.1">IsomorphismFpGroup</a> in the
<font face="Gill Sans,Helvetica,Arial">GAP</font> reference manual) to calculate a finitely presented group and
an isomorphism onto it, if some calculation does not work automatically.
<p>
<a name = "SSEC013.4"></a>
<li><code>Abelian Invariants</code>
<p>
computes and displays the abelian invariants of <i>u</i>.
<p>
<a name = "SSEC013.5"></a>
<li><code>Coset Table</code>
<p>
computes a coset table for <i>u</i>.
<p>
<a name = "SSEC013.6"></a>
<li><code>IsomorphismFpGroup</code>
<p>
computes a finitely presented group that is isomorphic to <i>u</i> and displays
the number of generators and relators of it.
<p>
<a name = "SSEC013.7"></a>
<li><code>Factor Fp Group</code>
<p>
computes the factor group of the whole group by <i>u</i>, if <i>u</i> is normal.
<p>
Note that the exact result of all these information displays is stored in
the global variable <code>LastResultOfInfoDisplay</code> after each operation. So you
can access this easily from the <font face="Gill Sans,Helvetica,Arial">GAP</font> command prompt. It is also returned
as <code>last</code> value.
<p>
<p>
[<a href = "chapters.htm">Up</a>] [<a href ="CHAP004.htm">Previous</a>] [<a href ="CHAP006.htm">Next</a>] [<a href = "theindex.htm">Index</a>]
<P>
<address>XGAP manual<br>Mai 2004
</address></body></html>
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