Finding out which subgroups a group contains is one way to understand its structure. For example, subgroups of Z6 are {0}, {0, 2, 4} and {0, 3}—trivial subgroups, multiples of 2 and multiples of 3. In the group D6rotations form a subgroup, but reflections do not. That’s because two reflections performed in sequence produce a rotation, not a reflection, just as adding two odd numbers produces an even number.
Certain types of subgroups called “normal” subgroups are especially useful to mathematicians. In a commutative group, all subgroups are normal, but this is not always true in general. These subgroups retain some of the most useful properties of commutability without forcing the entire group to commutate. If a list of regular subgroups can be determined, then the groups can be divided into components in the same way that integers are divided into products of prime numbers. Groups that have no regular subgroups are called simple groups and cannot be further divided, just as prime numbers cannot be factored. group ZN simply when N are prime—for example, multiples of 2 and 3 form internal normal subgroups Z6.
However, simple groups are not always so simple. “It’s the biggest misnomer in mathematics,” Hart said. In 1892, mathematician Otto Hölder suggest that researchers gather a complete list of all possible finite simple groups. (Infinite groups such as integers form their own field of study.)
It turns out that almost all finite simple groups look like this ZN (for prime values of N) or fall into one of two other families. And there are 26 exceptions, called discrete groups. Pinning them down and proving that there was no other possibility took more than a century.
The largest sporadic group, appropriately called the monster group, was discovered in 1973. It has more than 8×1054 elements and represents geometric rotations in a space of nearly 200,000 dimensions. “It’s crazy that people can find this,” Hart said.
By the 1980s, much of the work required by Hölder seemed to have been completed, but it was difficult to prove that there were no more sporadic groups left there. Classification was further delayed when in 1989 the community found gaps in an 800-page piece of evidence from the early 1980s. A new piece of evidence was finally published in 2004, completing the classification.
Many structures in modern mathematics—for example, rings, fields, and vector spaces—are created as more structures are added to groups. In rings you can multiply as well as add and subtract; In fields, you can also divide. But underneath all these more complex structures is still that original group idea, with its four axioms. “The richness that can be had in this structure, with these four rules, is amazing,” says Hart.
Original story reprinted with permission from Quanta Magazinean editorially independent publication of Simons Foundation has a mission to advance public understanding of science by addressing research developments and trends in mathematics, physics and the life sciences.
