from Niki Andersen

While working on the Mid-Term Exam for Math 817T this summer, I decided to get "lazy" and do a little bit of the work using Mathematica.  Dr. Fowler guided me to a web site for an Abstract Algebra Add-On for Mathematica.  I used this add on to create Cayley Tables. Here is an example of the Cayley table Dihedral 4.  (If you are unfamiliar with D4, it is the possible rotations and reflections of a square.)  In order to use an Add-On, you must load for each notebook that it is necessary.

The function above loaded the Abstract Algebra Add-On that I need to show you the rest.  

A question on our Mid-Term exam asked to find two isomorphisms, that is two groups that have the same "color scheme" and dimensions.  I decided to use Mathematica to illustrate this. You will notice that U[10] is isomorphic to Z[4].

You will see that the two Cayley Tables do not appear to be Isomorphic, but if the order of terms is changed in Z[4], you will quickly see the isomorphisms.  I had to form my own groupoid to change the order of the terms.  I called it G3.  When I want to make a Cayley Table of my new group, I can call it G3.

Now, if you compare the Cayley table of U[10] with that of G3, (which is actually Z[4]), you will see that their color schemes match, and that they are isomorphic.