PascGalois Zn Help: Introduction to the PascGalois Project


PascGalois is a project devoted to the visualization of concepts in Abstract Algebra. Through the use of automata, students are able to visualize some of the more difficult concepts in group theory. The images produced by this program and the recoloring schemes that are possible allow the student to visualize group properties as well as structure. For example, the images can clearly display a group's subgroups, quotient groups and with a little effort even subnormal series. The idea behind the images is an old concept with a new twist. Most mathematics students have seen Pascal's triangle and possibly even Pascal's triangle (mod n) where the entries from 0 to n - 1 are all colored a different color. For example, Pascal's triangle mod 2 looks like


and the Pascal triangles mod 5 and mod 6 are, respectively,


Notice in the triangle mod 6 there appears to be 2 superimposed images, one which is the mod 2 triangle and one that is the mod 3 triangle. These are, of course, the prime factors of 6 but they also represent the non-trivial subgroups of Z6. Thinking in terms of subgroups, if we draw the mod 2 and mod 3 triangles we see that inside the structure of Z6 we have things that look like Z2 and Z3. Hence Z6 has subgroups that are isomorphic to the groups Z2 and Z3.


This is where the new twist comes in. Why not try the same investigation with other finite groups? Take Zn X Zm, Dn or Sn, start with a seed row of elements from the group and construct the triangle in the same way as Pascal's triangle, except use the group's operation instead of addition. For example, triangles using the Quaternions and S3 are below.

The PascGalois JE program is capable of graphing both one and two dimensional automata over finite group structures. It uses a group operation engine capable of calculating arbitrary products and quotients of Zn, Zn (under multiplication), Dn, Sn, the Quaternions, Qn (Generalized Quaternions) and Cn (Dicyclic groups). It also has the capability of fully integrating user-defined structures. The interface was built to facilitate zooming and positioning, coloring of subsets, subgroups and cosets in order to extract more information about the group's structure quickly. It also has facilities for saving and loading workspace files as well as the saving of images and color/element correspondences. The program supports drag and drop of colors between two or more windows and or running applications to facilitate the investigation of isomorphisms. This version also has a group calculator that allows the user to calculate single results as well as generate subgroups and cosets. This program is an ongoing project and will evolve as the research in abstract algebra visualization evolves. The current version was written in Java so that it is platform independent.

The PascGalois Zn program is a special version of PascGalois JE which will graph cellular automata over just Zn. The difference between the Zn and JE versions is that the update rule for the Zn edition allows for both multiplication and addition where the JE edition allows only one operation at a tome. The Zn edition is also faster due to the fact that the group calculation engine is bypassed. The Zn edition has a slightly easier interface and will apply changes to color schemes and setup options automatically. So if you intend on investigating only Zn this is the program for you but if you are interested in other finite group structures you should be using the PascGalois JE program.

The creation of this program was made possible through a grant from the National Science Foundation (NSF #DUE-0339477).


Related Links: