Finding out what subgroups a group contains is one way to understand its structure. For example, subgroups of z6 are {0}, {0, 2, 4} and {0, 3}: the trivial subgroup, the multiples of 2 and the multiples of 3. In the group d6rotations form a subgroup, but reflections do not. This is because two reflections performed in sequence produce a rotation, not a reflection, just as the addition of two odd numbers results in 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 preserve some of the most useful properties of commutativity, 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 the same way that integers can be divided into products of prime numbers. Groups that do not have normal subgroups are called simple groups and cannot be decomposed further, just as prime numbers cannot be factored. the group znorth It’s simple only when north is prime: multiples of 2 and 3, for example, form normal subgroups in z6.
However, simple groups are not always so simple. “It’s the biggest misnomer in mathematics,” Hart said. In 1892, the mathematician Otto Hölder proposed that researchers gather a complete list of all possible simple finite groups. (Infinite groups, like integers, form their own field of study.)
It turns out that almost all simple finite groups look alike znorth (for prime values of north) or belong to one of the other two families. And there are 26 exceptions, called sporadic groups. Defining them and proving that there are no other possibilities took more than a century.
The largest sporadic group, aptly named the monster group, was discovered in 1973. It has more than 8×1054 elements and represents geometric rotations in a space with almost 200,000 dimensions. “It’s crazy that humans can find this,” Hart said.
By the 1980s, most of the work Hölder had asked for seemed to have been completed, but it was difficult to prove that there were no longer sporadic groups out there. Classification was further delayed when, in 1989, the community found gaps in an 800-page test from the early 1980s. a new test was finally published in 2004, finishing the classification.
Many structures in modern mathematics (rings, fields, and vector spaces, for example) are created when more structure is added to groups. In rings, you can multiply, as well as add and subtract; in fields, you can also split. But beneath all these more intricate structures lies the same original group idea, with its four axioms. “The wealth that is possible within this structure, with these four rules, is mind-blowing,” Hart said.
original story reprinted with permission of Quanta Magazinean editorially independent publication of the Simons Foundation whose mission is to improve public understanding of science by covering developments and trends in research in mathematics and the physical and biological sciences.