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Old group theory: How to classify all finite simple groups into four major categories?
In mathematics, the classification of finite simple groups (often called the "gigantic theorem") is an important result of group theory, which states that every finite simple group can be divided into four major categories: cyclic groups, alternating groups, Lie groups, or 26 special exceptions, which are called "occasional groups". These proofs spanned thousands of pages and hundreds of journal articles by approximately 100 authors, mostly published between 1955 and 2004.
Simple groups are considered the basic building blocks of all finite groups, just as prime numbers are the basic building blocks of the natural numbers.
The "Gigantic Theorem" is not only an important achievement in mathematical group theory, but also has wide applications in many branches of mathematics. Structural problems of simple groups are often reduced to problems about finite simple groups. Thanks to the classification theorem, we can solve the problem by examining only each family of simple groups and some occasional groups. Daniel Gorenstein announced in 1983 that finite simple groups had been completely classified, but due to his misunderstanding of some results, this announcement was actually premature. It was not until 2004 that Aschbach and Smith completed the proof of classification in a 1,221-page paper.
Summary of the classification theorem
The process of proposing a classification theorem is very long and tedious. The proof process can be divided into several main parts, especially the classification of groups of small 2nd order and component-type groups. The lower 2nd order of simple groups mainly includes some small-rank Lie groups and some alternating groups. The structural forms of these groups show the role that finite simple groups play in the beautiful structure of mathematics.
The classification of groups of small order 2, especially of order 2 or less, relies almost entirely on the theory of ordinary and modal roles, which is almost never used directly elsewhere in classification.
Another major classification direction is component groups. These groups have a structural correlation. By observing a certain centralizer, we can start the process of classification. We can understand the complexity of groups through the display of these correlations.
Characteristic Type 2 Groups and Existence
Regarding the characteristic type 2 groups, the classification of this part is equally important, especially the attribute analysis of all 2-local subgroups is the core. In the study of these groups, several results of Yalperin and Aschbach significantly advanced the classification process.
The classification theorem requires not only proving the existence of each simple group but also checking its uniqueness.
History and future prospects of classification
Historically, in 1972, Gorenstein proposed a plan to complete the classification of finite simple groups, which included a total of 16 steps. Each step represents an important theoretical cornerstone in group theory. Over time, second-generation classification proofs took shape, an innovative effort that helped simplify the cumbersome proofs of the past. Furthermore, this process demonstrates the evolving research methods in group theory.
New generations of proof work have made mathematicians more experienced, and the study of group theory has been enhanced by new techniques available to them.
In short, the classification of finite simple groups is a long-term and important topic in mathematics. From preliminary exploration to today's profound understanding, this process not only enriches the connotation of group theory, but also promotes the development of other fields of mathematics. Can future research provide more efficient classification methods? Is this a question worth thinking about for all mathematicians?
