UF Mathematics

About Group Theory

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Groups

A group is a set having a certain mathematical structure. Groups arise in many mathematical settings and in studies invoving symmetries in physics and chemistry. For example, groups are used in the classification of stereo-isomers: different molecules with the same chemical formula.

For a concrete example of a group, consider the symmetries of the square. Clockwise rotations I=R₀, R₉₀, R₁₈₀, R₂₇₀, of 0, 90, 180, and 270 degrees, and reflections V, H, about the vertical and horizontal axes, and reflections F₁, F₂ about the diagonals are the motions that carry a square back onto itself. Since any combination of these motions is again one of them, they form a group. Below a multicolored square is pictured with the effects of these motions on it.

I	R₉₀	R₁₈₀	R₂₇₀	V	H	F₁	F₂

Research in group theory at the University of Florida is carried out by mathematicians in Algebra Research Group.

The Inverse Galois Problem:

When Galois investigated the solvability of polynomial equations in terms of radicals, he studied certain types of groups associated with such equations which are called Galois groups today. Groups arise in a wide variety of settings in different ways. The Inverse Galois Problem concerns the question whether for any given group there exists a polynomial equation such that the Galois group associated with the solution of this equation is the given group. The conjecture is that is always the case, but the problem in full generality is still unsolved. Thompson has collaborated with mathematicians at the University of Florida on research related to the inverse Galois problem.

Simple Groups:

In any theory, one is interested in studying complex objects in terms of objects which are simpler in structure. For instance, in Number Theory, one looks at the decomposition of integers into products of prime numbers which have no positive divisors other than 1 and the (prime) number itself. Similarly, in group theory, one studies simple groups, those which have no normal subgroups except the identity and the whole group. For many years, the classification of all the finite simple groups was the most important problem in group theory. This classification was completed in the 1980's owing to the collective effort of many great mathematicians, including John Thompson.

Solvable Groups:

Another class of groups which are of importance are the solvable groups. A famous problem that remained unsolved for many years was the conjecture that every group with an odd number of elements is solvable. It was this ``odd order problem'' that Feit and Thompson solved in a paper running to 273 pages - an odd number of pages.

To learn more about groups and their applications, consider the following:

Wallpaper Groups (Plane Symmetry Groups) by David Joyce, Clark University
Reed-Solomon error correcting codes by Barry Cipra, SIAM News
based on looking at polynomials over finite fields
Quantum error correcting codes by Neil J. Sloane, AT&T Shannon Lab
An article in the Notices of the AMS December 2000 by Noam D. Elkies discusses linear codes and how some are related to sporadic simple groups
Crystallographic Topology at Oak Ridge National Laboratory
Symmetry and Complexity - Fundamental Concepts of Research in Chemistry, by Klaus Mainzer, Int'l J. for the Philosophy of Chemistry, 1997.

Thompson and National Medal of Science * University of Florida * Mathematics * Contact Info

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This page was last modified on November 15, 2000.