Guide to Group Analyzer (version 5)
Paul G. Glenn
Department of Mathematics
Catholic University of America
Washington, DC 20064 
Phone: 202-319-5221, E-mail: glenn@cua.edu

What is Group Analyzer (GA) ?

Group Analyzer is a program for students and instructors of
elementary group theory. It enables its users to perform various
finite group computations in a calculator-like manner with groups of
order up to 400. They can find various subgroups and quotient groups,
and in general experiment with the group. For students, the goal of
such calculations, performed in many different groups, is to get a
better grasp of the basic concepts.

Here is how GA works:

1.  After starting GA itself, the user loads in the information
    for a group contained in a separate {\em group data file}. (Some
    samples are provided in the ``package'' and there are separate programs
    for creating others. These are also part of the package.)

2.  The user issues a command for some calculation. This is
    equivalent to pressing a button on a calculator. Commands can be
    given from menus, by clicking on boxes in the ``controls'' window or
    by keyboard equivalents.

    Some commands require no input (e.g., the command to display the
    elements of the group) while others may require one or two subsets of
    the group as input. The output of a calculation (one or more subsets
    of the group) appears in the GA output window.

3.  The output can be manipulated in a variety of ways. First, the
    output window (which is scrollable) contains plain text that one may
    select, copy and paste into other applications. A user could cut and
    paste to build a separate record of his or her calculations. Second,
    one may instruct GA to keep an annotated journal of all the
    results obtained. This journal is a separate plain-text file that one
    may examine and edit later on.

4.  At any time, the user can study groups related to the group given
    by the group data file without loading in a new data file. If G
    denotes the original group, if H is any subgroup of G and if N
    is any normal subgroup of H, then one can set GA to work with
    the quotient group H/N. This is called the ambient group.

    GA can also create separate group data files for any H/N for separate
    analysis.

5.  At any time, the user can load in a new group data file.


Group Analyzer's role in an algebra course:

Students work, typically, with two types of examples. One type
involves very small order groups in which explicit calculations are
easy for beginners. A second type, requiring familiarity with the
basics and some experience, involves larger groups whose properties
and characteristics are found using theorems and ad hoc indirect
reasoning. With this type of example, one would almost certainly 
not want to hand-calculate lists of elements such as cosets, the set of
commutators, etc.

One purpose of GA is to make feasible another approach to examples.
As with very small groups, the students work directly with an
explicit complete specification of the group's operation (in the form
of a table stored in the computer's memory). However, the groups are
of ``intermediate'' size, of order up to 400, and are big enough to
illustrate various ideas of finite groups in non-trivial ways. With
such examples, students can illustrate or conjecture theorems, search
for counter-examples and check assertions based on indirect
reasoning. Thus the software serves as an adjunct to indirect
reasoning. At the same time, it allows students to work with larger
groups in an explicit way similar to their experience with very small
groups.

Availability:

Group Analyzer and the auxiliary programs are free to
the academic community. The package is available at the Mathematics
Software Archive ( http://archives.math.utk.edu/ ).