Determining what subgroups a group contains is one way to understand its structure. For example, subgroups WITH6 are {0}, {0, 2, 4} and {0, 3} – negligible subgroup, multiples of 2 and multiples of 3. In the group D6rotations form a subgroup, but reflections do not. This is because two reflections taken in succession result in a rotation, not a reflection, just as adding two odd numbers produces an even number.
Certain types of subgroups, called “normal” subgroups, are particularly useful to mathematicians. In an alternating group, all subgroups are normal, but this is not always true more generally. These subgroups retain some of the most useful properties of commutativeness without forcing the entire group to be commutative. If a list of normal subgroups can be identified, the groups can be divided into components in much the same way that integers can be divided into products of prime numbers. Groups that have no normal subgroups are called basic groups and cannot be further divided any more than prime numbers can be factorized. Group WITHN It’s only basic when N is a prime number – for example, multiples of 2 and 3 form normal subgroups of w WITH6.
However, basic groups are not always that basic. “It’s the biggest misnomer in mathematics,” Hart said. In 1892, the mathematician Otto Hölder he suggested that researchers gather together complete list of all possible finite basic groups. (Infinite groups such as integers form their own field of study.)
It turns out that almost all finite groups are basic or look like WITHN (for prime values N) or belong to one of two other families. There are 26 exceptions, called sporadic groups. It took over a hundred years to immobilize them and show them that there were no other options.
The largest sporadic group, aptly called the monster group, was discovered in 1973. That’s what happened more than 8×1054 elements and represents geometric rotations in a space with almost 200,000 dimensions. “It’s just crazy that people could find something like this,” Hart said.
By the 1980s, most of the work Hölder called for seemed to have been completed, but it was hard to demonstrate that there were no longer sporadic groups there. Classification was further delayed when, in 1989, the community found gaps in one 800-page evidence from the early 1980s. New evidence has finally been published in 2004, completing the classification.
Many structures in contemporary mathematics – such as rings, fields, and vector spaces – arise when more structures are added to groups. In rings you can multiply, as well as add and subtract; you can also divide in the fields. But beneath all these more complicated structures lies the same original idea of the group, with its four axioms. “The richness that is possible in this framework, with these four principles, is staggering,” Hart said.
Original story reprinted with permission Quanta Magazineeditorially independent publication Simons Foundation whose mission is to escalate society’s understanding of science by incorporating research developments and trends in mathematics and the physical and life sciences.