Bunch-Hellemans, Galois and group theory

Popular legend has it that 20-year-old Evariste Galois, thinking he might be killed in a duel over a woman’s honor, invented group theory –– one of the most basic and important concepts of modem mathematics –– on the night of May 29,1832. Galois used his new concept to prove that equations of the fifth degree –– quintics –– and higher could never be solved. He was indeed killed the following day, but because his ideas were so radical it took 150 years for mathematicians to work out their implications.

It is a dramatic legend, but almost entirely untrue. Paolo Ruffini proved that the general quintic could not be solved by algebraic means before Galois was born, and independently Niels Abel gave an even better proof in 1824. Although Galois contributed to group theory he cannot lay claim to being its inventor. The notes he wrote out that night were not new mathematics; over the previous five years Galois had submitted three versions of the same material to the French Academy, but two great mathematicians had managed to ignore or misplace the first two versions, and the third was turned back for lack of clarity; it is even unlikely that the duel was over a woman’s honor.

Group theory can be traced back to origins in geometry, number theory, and permutations, as well as the algebraic studies of Ruffini, Abel, and Galois, although it was Galois who first used the term group. Possibly the earliest use of group theoretic ideas was by Leonhard Euler in 1761 in a paper on number theory, although Euler did not explicitly label the concepts as anything special. Later, a group was defined as a set of objects that obeys four simple rules for a given operation. For the mathematically inclined, groups are closed and associative and possess an identity element and inverses with respect to it: Closed means that when two elements of a group are combined by the group operation, another element of the group always results. Associative means that three elements can be combined starting with either the first pair or the second pair of the three and the result does not change. The identity element is a member of the group that does not change an element combined with it. Inverses are pairs of elements that when combined produce the identity element.

An example of a group is the positive and negative integers where the operation is addition and the identity element is zero. A finite group might consist of all the transformations that bring a capital letter of the alphabet, other than O, into itself: A remains the same when reflected, I or H when reflected or turned 180°, N when turned but not when reflected, etc.

Galois’ accomplishment, while misrepresented in popular legend, was quite real. Nearly 20 years after his death, the concepts of group theory had advanced enough so that his obscurely written paper could be clearly understood. He had developed a general method for determining the solvability of equations, going considerably beyond Ruffini and Abel. Along the way, he had independently created large chunks of group theory.

Group theory came to pervade all of mathematics. In 1872 Felix Klein suggested that different types of geometry be defined in terms of groups in his famous Erlangen Program. This was largely accomplished in the late 19th and early 20th centuries. In fact, a purely geometric concept, such as topology (the branch of geometry concerned with connections rather than shapes), evolved to produce a purely algebraic theory of topology based on groups.

Modern algebra books introduce groups in the early chapters and use the concept thereafter. Even branches of mathematics with no explicit use of group theory are structured in ways that emerged from the theory. One textbook that addresses all of classical mathematics states in its introduction, “If a person is not ready and anxious to explore the concept of a group further, we recommend that he reconsider his relationship to mathematics.” Despite its pervasive influence in mathematics, group theory was not seen as particularly useful in applications until recently.

Although it was suggested as a tool in chemistry, nothing much emerged. Crystallographers were the main beneficiaries outside of mathematics. Then, in 1926, Eugene Paul Wigner introduced groups to particle physics. After World War II, a number of physicists began to think about how group theory could be used. In the early 1960s Sheldon Glashow, Murray Gell-Mann, and others discovered that group theory enabled them to predict the properties of particles before the particles were observed and to make sense out of the many new particles that were being discovered.

By the 1980s a “standard model” of particle physics had emerged based on group theory, and physicists were using groups as a tool in trying to develop grand unified theories (GUTs) that combine all the known forces except gravity. The even more ambitious superstring theory, which includes gravity, is also based on groups.

Meanwhile, the very mathematicians who specialized in one part of pure group theory worked themselves out of a job. It had become apparent that there were only a finite number of simple groups that have a finite number of members. (Roughly speaking a simple group is one without a certain class of subgroups.) Group theorists doggedly worked out each possible group in succession. Finally, in 1980, Robert L. Griess, Jr. worked out the last possible finite group, nicknamed the “monster” because it is so large.