Knowing what subgroups a group contains is one way to understand its structure. For example, the following subgroup is Z6 {0}, {0, 2, 4}, and {0, 3} are trivial subgroups, multiples of 2, and multiples of 3. within the group D6rotation forms a subgroup, but reflection does 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 determine 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. Groups that have no regular subgroups are called simple groups and cannot be further factorized, just as prime numbers cannot be factorized. group Zn It is simple only if n is a prime number. For example, multiples of 2 and 3 form a normal subgroup. 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 get together 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 look like one of the following: Zn (For prime numbers n), or 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 sporadic group, called the Monster Group, was discovered in 1973. 8×10 or more54 Use elements to represent geometric rotations in a space of approximately 200,000 dimensions. “It’s really crazy that this was discovered by humans,” Hart said.
By the 1980s, it appeared 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. new evidence finally published Classification ended in 2004.
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. “The richness that is possible within this structure with these four rules is amazing,” Hart says.
original story Reprinted with permission from Quanta Magazineis an editorially independent publication. simmons foundation Its mission is to enhance the public’s understanding of science by covering research developments and trends in mathematics, physical sciences, and life sciences.
