> < ^ Date: Thu, 21 Mar 1996 19:40:48 +0100
> < ^ From: Franz Gaehler <gaehler@itap.physik.uni-stuttgart.de >
^ Subject: Graphic Conjugacy Class Lattice

Dear GAP-Forum,

I have uploaded my extensions of XGAP routine GraphicLattice
to ftp://ftp.math.rwth-aachen.de/pub/incoming/glatcc-1.0.tar.gz.
GraphicLattice now accepts an additional argument "Conjugacy Classes".
This argument instructs GraphicLattice to display a graphic
"subgroup conjugacy class lattice". Please note that the current
version is in file


The files glat.doc, glatcc.tar.gz and glatcc.zoo belong to an earlier
version. That earlier version works, but is less complete.

Below I have attached a short documentation.


With kind regards,

Franz G"ahler

Graphic Lattice for Conjugacy Classes of Subgroups - Documentation

This package is an extension to XGAP Version 1.3. It provides a
new version of GraphicLattice, which accepts a further, optional
argument "conjugacy classes" (or "Conjugacy Classes"). This argument
can also be abbreviated.

GraphicLattice( <group>, "conjugacy classes" ) returns a graph whose
vertices represent conjugacy classes of subgroups of <group>. Two
Vertices are connected if the representatives of one of the conjugacy
classes contain maximal subgroups from the other conjugacy class.

Compared to GraphicLattice( <group> ), the resulting graph is often
greatly simplified, as subgroups in the same conjugacy class are merged
into a single vertex. Of course, there is also some loss of information.
However, one can still inquire about all properties which are class
functions, so that a lot of useful information is still available.

If the mouse points to a vertex, the right mouse button menu allows to
inquire about properties of the representative of the conjugacy class
of a vertex. The results are all class functions, and thus are valid
throughout the whole class. In addition to the menu entries from
GraphicLattice( <group> ), an entry "Class size" has been added.

The Subgroups menu contains essentially the entries of the Subgroups
menu of GraphicLattice( <group> ). These entries still have the same
meaning, except that the conjugacy classes of the corresponding
subgroups are returned, instead of the subgroups themselves.
The menu entries "Closure" and "Closures" return the closures of
the whole subgroup conjugacy class, which is equal to the normal
closure of the selected representatives. Similarly, "Intersection"
and "Intersections" return the intersections of entire classes,
which is equal to the core (in the top level group) of the selected
representatives. "CommutatorSubgroups" has been omitted, as there
does not seem to be an obvious definition for this. Finally, a new
entry "Number of Copies" has been added to the Subgroups menu.
For each pair in the set of selected vertices, a message is issued,
telling how many maximal subgroups from one class are contained
in a representative of the other class, and how many minimal supergroups
a representative of one class has in the other class. If these numbers
are zero, no message is issued.

The CleanUp menu has been updated as well. "Select Representatives"
has been removed (this is not very useful, if every vertex is a
representative), and "Rotate Conjugates" (which isn't useful either
here) has been replaced by "Swap Vertices". This latter option allows
to swap two vertices, provided they are in the same horizontal strip.

All other menus work as for GraphicLattice( <group> ).

A vertex is represented internally by the representative of its conjugacy
class, and not by the conjugacy class itself. The benefit of this is
twofold: we can use most of the entries of the right mouse button menu
without modification, and Selected( <sheet> ) returns a list of subgroups,
which can be fed directly into InteractiveLattice. This allows to analyze
a subgraph in more detail, including the conjugates.

Franz G"ahler <gaehler@orphee.polytechnique.fr>
Centre de Physique Theorique, Ecole Polytechnique, F-91128 Palaiseau, France

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