Knowing what subgroups a group contains is one way to understand its structure. For example, the subgroups of Z6 are {0}, {0, 2, 4}, and {0, 3}. These are trivial subgroups, multiples of 2, and multiples of 3. In group D6, rotations form a subgroup. , but the reflection is not. This is because, just as adding two odd numbers yields an even number, performing two reflections in sequence produces a rotation rather than a reflection.
A particular type of subgroup, called a “normal” subgroup, is particularly useful to mathematicians. In commutative groups, all subgroups are normal, but this is not necessarily the case more generally. These subgroups retain some of the most useful properties of commutativity without forcing the entire group to be commutative. Once we can identify the list of regular subgroups, we can divide the group into its components, much like we would divide an integer into a product of prime numbers. A group that has no regular subgroups is called a simple group, and it cannot be further factorized, just as a prime number cannot be factorized. The group Zn is simple if and only if n is prime. For example, multiples of 2 and 3 form a normal subgroup of Z6.
However, a simple group is not necessarily simple. “This is the biggest misnomer in mathematics,” Hart says. In 1892, mathematician Otto Herder proposed that researchers create a complete list of all possible finite simple groups. (Infinite groups such as the integers form their own field of study.)
It turns out that almost all finite simple groups either look like Zn (a prime number of n) or fall into one of the other two families. And there are 26 exceptions called the sporadic group. It took more than a century to track them down and show that there were no other possibilities.
The largest scattered group, called the Monster Group, was discovered in 1973. This group has at least 8 × 1054 elements and represents a geometric rotation in approximately 200,000 dimensions of space. “It’s really crazy that this was discovered by humans,” Hart said.
By the 1980s, it seemed that much of the work Herder had called for had been completed, but it was hard to see that there were no more sporadic groups remaining. Classification was further delayed in 1989 when the community discovered a gap in the 800-page proof from the early 1980s. A new proof was finally published in 2004, completing the classification.
Many structures in modern mathematics (e.g. rings, fields, vector spaces) are created when more structures are added to a group. Rings allow addition and subtraction as well as multiplication. You can also split by field. But underneath all these more complex structures is the same original group concept with four axioms. “With these four rules, the richness that is possible within this structure is amazing,” Hart says.
Original story with permission from Quanta Magazine, an editorially independent publication of the Simons Foundation whose mission is to advance public understanding of science by highlighting research developments and trends in mathematics and the physical and life sciences. It is reprinted here.
