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