Group Analyzer page

Dr. Paul G. Glenn
Department of Mathematics
Catholic University of America

Abstract for Group Analyzer

HTML version of the User's Guide

System and Hardware requirements

Individual files for the Group Analyzer package

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 examine 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.

Students work, typically, with two types of finite 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 experiistics are found using theorems and ad hoc mathematical 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 Group Analyzer is to make feasible another approach to finite 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 arrived at using (or suggested by) 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.

I have tried to design Group Analyzer so that it will be easy for students to use (and learn to use) and not become a distraction from the main task of learning group theory. I hope that its use will help students develop their mathematical reasoning. It should allow students to experiment with real examples without calling excessive attention to itself and (I hope) without doing so much of the work as to make the user a mere spectator.

Group-structure calculations performed by Group Analyzer:

In all cases, the input (if required) is user-chosen: either one or two subsets of the group.

Subgroup generated by a subset of the group

Multiplication of subsets A and B

Inverses of the elements of a subset A

n'th power of each element of a subset. (n chosen by user)

Conjugate of subset A by subset B

(Conjugates each element of A by each element of B)

Conjugacy class of each element of a subset

Centralizer of a subset

Normalizer of a subset

Normal closure of the subgroup generated by a subset

Sets of commutators

The set of all commutators [a,b] where a belongs to A, b belongs to B.

Commutator subset of the group

Union, intersection, difference of two subsets

Left cosets of a selected subgroup

Left coset representatives of a designated subgroup.

Center of the group

Order n elements of the group. (n chosen by user).

Conjugacy classes of the group

Conjugacy class representatives

Group Analyzer can also perform calculations in homomorphic images of any subgroup of the given group. In particular, it can calculate the image of a subset S of G in the quotient group G/N and can calculate the inverse image in G of a subset of G/N.

The Group Analyzer applications, the group data file maker applications and the documentation are copyright 1992-1999, all rights reserved, by Paul G. Glenn. They are offered FREE to the academic community. The package may be distributed provided it is done so intact and whole and free of charge.

Group Analyzer runs on Macintosh computers under System 7.x and MacOS 8.x. There are two versions: one for 68k machines (called GA 68k) which also runs on PPC machines under emulation; the other is a Power Mac version (called GA ppc) which runs on Power Macs only. The two versions function the same mathematically. The Power Mac version performs certain operations somewhat faster. Both versions require about 750k of memory.

All the files below are compressed (using Compact Pro) and binhex'd. They may be opened with Stuffit Expander.

The Group Analyzer applications (Size: 148K)

This file contains the PPC and 68k versions of the application. These versions: Summer, 1999.

This file contains a sampling of group data files (the files which contain data for the individual groups to be loaded into Group Analyzer). The next file contains special-purpose applications for creating group data files. This version: September, 1999.

This file contains small special-purpose applications for creating group data files. This version: September, 1999.

This file contains various documents related to Group Analyzer. These are:

• User's Guide: This gives a complete description of Group Analyzer's operation. It also includes a "question/answer" section to give brief answers to basic questions which may help users get started with Group Analyzer quickly. Format: LaTeX dvi file. Current version: September, 1999.
• Tutorial: This runs through the individual steps of a "session" with Group Analyzer illustrating some of its operations. It's designed so that students can learn the essentials early in the semester. Format: LaTeX dvi file. Current version: September, 1999.
• Sample activities: These are worksheets which lead students through some calculations with groups. Each worksheet focusses on a particular idea. Teachers are welcome to use these worksheets as they are or in modified form. Format: LaTeX 3 source files (plain text). No special macros or style files are required. Current version: September, 1999.