ITC stands for Interactive Todd Coxeter, it is a program that allows you to execute interactively single steps in an enumeration of the cosets of a subgroup of a finitely presented group using the graphics surface XGAP of GAP and thus to see in various windows exactly what is happening. In this chapter we will
Coset Enumeration, in the sequel abbreviated as CE, also known as the Todd-Coxeter procedure, is not only the oldest but also one of the most often used tools in Computational Group Theory and also probably the most often implemented one.
An easy introduction to CE and some of its variants is given in Neu82. In his book Sims94 Charles Sims puts CE into a much more general context based on the theory of automata. Both texts provide a fairly comprehensive guide to the extensive literature on CE. So we will assume in the sequel that the user of this manual understands the basic ideas of CE. We will use as far as possible the terminology of Neu82.
The main GAP library does contain routines for executing coset enumerations (see Section Coset Tables and Coset Enumeration in the GAP reference manual). These routines are partly written in the GAP language but for efficiency also use GAP kernel functions written in C. Some of these programs are used by ITC.
The most efficient implementation of CE, allowing many ``strategies'', at present is the package ACE by George Havas and Colin Ramsey (see Ram99) which is also available as a GAP4 package.
It should be understood that ITC is no competitor for ACE. ACE can successfully deal with coset enumerations with millions of cosets. On the other hand handling coset tables graphically on the screen, even with using scrolling, in practice limits the use of ITC to tables with a few thousand rows. For that reason also the speed of the single step in the enumeration does not matter as much and ITC therefore is not tuned for speed in the way ACE is.
In texts on CE usually examples are demonstrated, and whenever one teaches the method in class, one will have to work through some examples. In printing as well as on the blackboard this suffers from a certain difficulty: In CE a number of tables, the ``coset table'', the ``subgroup tables'', and the ``relation tables'' are being changed all the time, one has to gain information from (the closing of rows of) subgroup and relation tables and has to transfer this information into all relevant places of all tables. Sometimes, when so-called ``coincidences'' occur, one even has to erase and modify already existing entries in tables. Describing such steps in some detail as well as the ever changing tables soon occupy many pages in printing. Doing them in front of a class one can pretty soon get confused, forgetting to make entries or changes etc.
So it is a natural demand to have a program that allows one to perform such steps in windows on the screen of a computer where such changes can be cleanly made, are transferred to all relevant places in all tables automatically, and are marked by colours so that one can really ``play'' through nontrivial examples. This is what ITC provides. So in a first place it can be considered as a teaching tool.
However we hope that ITC can also be used for experiments that ultimately may lead to a better understanding of existing methods or even the design of new ones.
As explained in many papers about CE, the sequence in which steps are made has great influence on the efficiency of the methods. To be more exact: CE is basically a method of trial and error in which one defines cosets of a subgroup of a finitely presented group by constructing coset representatives as words in the generators of the group. But only the termination of the method (the closing of the tables) in the end guarantees that one has not made a mistake by defining more than one word representing the same coset. The discovery of such a mistake and its removal is what is called ``handling a coincidence'' and it is the number of occurrences of such mistakes (the number of coincidences encountered) that may vary immensely depending on the chosen sequence of definition of pretended representatives.
Therefore many different ``strategies'' for CE have been discussed in the literature, and programs such as ACE allow one to use a wide variety of such strategies. However in spite of very detailed case studies, in particular by George Havas, that lead to certain ``rules of thumb'', no strategy is uniformly the best one, a strategy that is good for one presentation may turn out to be much less good for another one (see for instance CDHW73, Hav91, HR99a, and HR99b). The possibility to try one's own strategy stepwise interactively therefore can be used to get some more insight into CE which to some extent is an art rather than a science.
Already in the building of ITC a nice side product was obtained
from such experimentation: It is important to understand that in many
cases coincidences cannot be completely avoided for a CE to terminate.
However when many coincidences have occurred, one is led to ask if
there is not a shorter sequence of definitions leading to the closure
of the tables. Starting from a given CE with coincidences involved,
one may try to ``prune'' the definition sequence from definitions that
in retrospect can be recognized as being redundant. Such a procedure
has for instance been employed many years ago ``by hand'' by John Leech
in a search for a short definition sequence for the Fibonacci Group
F(2,7) that we will take as one of our examples. A very nice
discussion of this idea and its history is given in Margaret Edeson's
Master's Thesis (see Ede89). Using ITC such ``pruning'' was
first done interactively. This led to an algorithm that is described
in the section ``The Short-cut Method'' (see Section The Short-cut Method). It is now part of ITC and can be called by
(see short-cut). In fact with the above mentioned old challenge
problem of F(2,7), it beats the formerly best known definition
sequence leading to closure of the tables.
Since ITC will store sequences of definitions we intend in the future to investigate the possibility to improve runs of a ``Modified TC'' (MTC) by using such ``pruned'' definition sequences.
ITC is written using the graphics facilities provided by the GAP4 package XGAP. We refer to the XGAP manual for a general description of these.
The ITC Coset Table as well as the Relation Tables and Subgroup Tables can be depicted in windows, and definitions be made in these by mouse click. Moreover tables with the sequence of definitions, of pending coincidences, and of gaps of length 1 in Subgroup or Relation Tables can be inspected in further windows and be used for deciding about the next step. A number of menus allow certain actions to be called.
In addition to the Coset Table Window which plays a special role, three different kinds of windows are used in ITC:
change default table size(see change default table size),
extend table size(see extend table size); in the menu of
read definitions from file(see read definitions from file),
write definitions to file(see write definitions to file), and
write standardized table to file(see write standardized table to file). Also such query windows are opened by the following ``bottom buttons'' of the Coset Table Window:
scroll by(see scroll by),
scroll to(see scroll to),
fill gaps(see fill gaps),
fill rows(see fill rows),
back to(see back to),
short-cut(see short-cut), and
mark cosets(see mark cosets).
Sheetthat via a pull-down menu allows one either to write the contents of the display window to a PostScript file or to close this display window. See the paragraph Button
Sheetin Section The Top Buttons. A display window is opened by clicking the menu entry
show current settings(see show current settings) in the menu opened via the top button
Settings. Further display windows are opened by clicking the following bottom buttons in the Coset Table Window:
show rels(see show rels),
show subgrp(see show subgrp),
show defs(see show defs),
show gaps(see show gaps), and
show coincs(see show coincs). Further display windows can be opened by clicking with the right mouse button: coset numbers or dots in the Coset Table (see The Coset Table), entries in the Table of Definitions (see The Table of Definitions), entries in the List of Length 1 Gaps (see The List of Length 1 Gaps), and entries in the List of Pending Coincidences (see The List of Pending Coincidences). Also the Subgroup Tables (see The Subgroup Tables) and the Relation Tables (see The Relation Tables) are display windows that can be opened by clicking the respective entries in the List of Subgroup Generators (see The List of Subgroup Generators) or the List of Relators (see The List of Relators), respectively. The windows of Relation Tables and of Subgroup Tables differ from the other display windows in a technical way: Like the Coset Table Window, they stay open while the contents of the displayed tables changes. Moreover, the Relation Tables are scrolled parallel to the Coset Table (see The Relation Tables).
Depending on the window system and window manager you use, placing a new window on your screen might be done automatically or might require you to use the mouse to choose a position for the window and pressing the left mouse button to place the window.
Making a window active is system and window manager dependent, in most cases you have either to move the pointer inside the window or you have to click the title bar of the window.
We recommend to switch on title bars for all windows in your window
manager setup. This seems to avoid a small bug in at least one window
manager we know of and makes it easier to move your windows
around. For the
twm window manager for example you can achieve this
with the option
If you do not want to do this and experience problems with small
dialog boxes that do not accept text entered via the keyboard, try to
start up XGAP with the command line option
-W. This invokes some
code within XGAP which works around a bug in some window managers.
Please note that because of some 16 bit limitations in the underlying X toolkit the present implementation of XGAP may not be able to handle very large virtual windows properly. For instance, sheets containing more than 1630 coset definitions may be corrupt.
In the following chapters we will describe the possibilities to work with windows in a systematic way and by way of examples of the use of ITC.
We thank Max Neunhöffer for providing the needed environment in his package XGAP and Thomas Breuer for frequent advice. A first version of ITC was implemented, still under GAP 3, by Ludger Hippe as part of his ``Staatsexamensarbeit'' in 1997.
Authors' e-mail addresses:
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