‘Groups’ Underpin Modern Math. Here’s How It Works

Figuring out which subgroups comprise a cluster is one way to understand its structure. For example, subgroups of Z₆ is {0}, {0, 2, 4} and {0, 3}—the smallest subgroup, multiple of 2, and multiple of 3. In the group D₆rotation creates a small group, but reflection does not. That’s because two consecutive reflections produce a rotation, not a reflection, just as adding odd numbers results in one.

Certain types of subgroups called “normal” subgroups are especially useful for mathematicians. In a dynamic group, all subgroups are common, but this is not always true in general. These groups retain some of the most useful features of the exchange, without forcing the entire group to change. If a list of common subsets can be identified, the groups can be divided into parts in the same way that numbers can be divided into products of primes. Groups without regular subgroups are called simple groups and cannot be further divided, since the prime numbers are incommensurable. The group Z_n is simple only if n is prime—multiples of 2 and 3, for example, form common subgroups in between Z₆.

However, simple groups are not always so simple. “It’s the biggest misnomer in math,” Hart said. In 1892, mathematician Otto Hölder proposed that researchers compile a complete list of all simple finite groups. (Infinite groups such as whole numbers form their own field of study.)

It turns out that almost all simple finite groups look like this Z_n (with significant values ​​of n) or fall into one of the other two families. And there are 26 exceptions, called sporadic groups. Pinning them down, and showing that there are no other possibilities, took a century.

The largest unusual group, aptly named the group of monsters, was discovered in 1973. It is more than 8 × 10.⁵⁴ elements and represents a geometrical rotation in space with approximately 200,000 dimensions. “It’s just crazy that people can find this stuff,” said Hart.

By the 1980s, much of the work that Hölder had requested seemed to have been completed, but it was difficult to demonstrate that there were no longer lasting groups. The classification was further delayed when, in 1989, the public discovered gaps in another 800-page document from the early 1980s. New evidence was finally published in 2004, ending the split.

Many modern mathematical structures—rings, fields, and vector spaces, for example—were formed when additional structure was added to groups. In rings, you can multiply and add and subtract; fields, you can also divide. But underneath all these complex structures is that original group theory, and its four axioms. “The richness that is possible within this structure, with these four rules, is impressive,” Hart said.

The first story reprinted with permission from Quanta Magazine, the independent editorial publication of Simons Foundation whose mission is to improve public understanding of science by incorporating research developments and trends in mathematics and the natural and biological sciences.