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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

glatcc-1.0.tar.gz.

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.

Enjoy!

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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