Handout

Finite Group Behavior:
Windows Software for Teaching
Beginning Group Theory
Edward C. Keppelmann University of Nevada, Reno
Ellen J. Maycock AMS and DePauw University
June 26, 2006
1. Introduction
The software Finite Group Behavior (FGB) is an easy-to-use, free, Windowsbased
program, designed not only to help beginning students learn about group
theory, but also to encourage more advanced students to pursue independent
research on …nite groups.
Finite Group Behavior was developed by Mr. Bayard Webb, who was a
student of Professor Edward Keppelmann during the early 1990’s. Ed Keppelmann
has continued to improve the software over the past decade. You will be
able to use a new super-version of FGB, completed for this minicourse. Although
Bayard Webb is no longer working on the software, he wished that the software
remain available free of charge to the mathematics community. The software
Exploring Small Groups, written by Ladnor Geissinger, was the inspiration
for Mr. Webb’s work. This laboratory is based on one contained in Laboratory
Experiences in Group Theory by Ellen Maycock Parker, published by the
Mathematical Association of America.
1

2. Endomorphisms
You can use FGB to construct homomorphisms from one group to another.
For the next two labs, however, we will focus on endomorphisms of a speci…c
group. Recall the following de…nitions:
De…nition 1. Let G and K be groups. A function T : G ! K is called a group
homomorphism if T (xy) = T (x)T (y) for every x; y 2 G: When K = G; we say
that T is a group endomorphism.
De…nition 2. The kernel of a homomorphism is the set
ker(T ) = fx 2 G j T (x) = 1 G g
De…nition 3. The image of a homomorphism T is the set
im(T ) = fy 2 K j 9x 2 G with T (x) = yg
We will work through an example using D 3 before you begin the rest of the
lab.
The computer will use the following two rules to compute the endomorphisms,
once you choose some initial values for the function:
1. i. Power rule: If you set T (a) = b, the computer calculates that T (a r ) =
b r :
ii. Homomorphism rule: If you set T (a 1 ) = b 1 and T (a 2 ) = b 2 , the
computer calculates that T (a 1 a 2 ) = b 1 b 2 :
Of course, the power rule is a special case of the homomorphism rule. It helps
students to see these spelled out, and it is then easy to deduce that the order of
b must divided the order of a when T (a) = b:
2.1. Example
Retrieve group D 3 or rename the elements by the following table:
1 A B C D E
1 r1 r2 m1 m2 m3

Choose the tab Homomorphisms, and choose as the Range 0602 or D 3
(if you have saved the table under that name).
We will create an endomorphism using the dialog boxes. Enter r1 in the
Assign box, and r2 in the To box. Click the To button. This command
will generate the images of 1; r1; and r2: With each question that pops up,
click OK, or All to streamline the process. To complete the assignments,
enter m1 in the Assign box and m1 in the To box, and click To. You
should have generated the full assignment, and the top two rows of the
table in the computer screen should match the following chart.
Element x 1 r1 r2 m1 m3 m2
Image T (x) 1 r2 r1 m1 m2 m3
What are the image and kernel of this endomorphism
As a second example, try mapping r1 to r1 and m1 to 1.
this time What would happen if you mapped r1 to m1
2.2. Further Work on Endomorphisms
What happened
1. Find three nontrivial endomorphisms of D 3 which are not automorphisms.
Write down the image and kernel of each. How many automorphism of D 3
are there Explain.
2. There are eight automorphisms of D 4 : Explain why and …nd two of them.
There are 27 nontrivial endomorphisms of D 4 . Find one whose image is
isomorphic to Z 2 and one whose image is larger than Z 2 . Write down the
kernel and image of each.
3. Make a conjecture about the number of automorphisms of D n :

3. Group Library of FGB, orders 3 - 16
The groups contained in the folders of various orders agree as much as possible
with the group library of ESG. You will probably want to rearrange some of the
folders, or create new ones, in ways that work best with your own materials. For
example, you may wish to have folders that contain all the groups, including the
cyclic ones, of each order. Or you may wish to have all the dihedral or dicyclic
groups together.
Below is the listing of the group library of all groups up through order 16.
The four-digit codes refer to group tables. The …rst two digits of the code give
the size of the group. Common names are given with most of the groups.

4. Information
4.1. Addresses
Edward C. Keppelmann
University of Nevada
Department of Mathematics & Statistics
AB601 MS084
Reno, NV 89557-0045
Phone: 775-784-6773 (o¢ ce)
Fax: 775-784-6378,
Email: keppelma@unr.edu
Url: http://unr.edu/homepage/keppelma/
Ellen J. Maycock
American Mathematical Society
201 Charles Street
Providence, RI 02904
Phone: 401-455-4101
Fax: 401-455-4004
Email: ejm@ams.org
4.2. Materials
The software Finite Group Behavior can be downloaded from the web
site: http://unr.edu/homepage/keppelma/fgb.html. The software is free,
although registration is required. Ed Keppelmann continues to improve
and expand the program, and welcomes suggestions.
The lab manual Laboratory Experiences in Group Theory, bundled
with the software Exploring Small Groups, is still available through the
MAA bookstore, online at
https://enterprise.maa.org/ecomtpro/Timssnet/products/TNT_products.cfm.
The labs in this manual can be adapted for use with FGB.