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Incredibly long math: Why does the proof of finite simple groups require a 100,000-page paper?
In the history of mathematics, the classification theorem of finite simple groups is widely called the "huge theorem". Its appearance has brought about a considerable revolution in the development of group theory. The theorem states that all finite simple groups are either cyclic or alternating, or belong to a broad infinite class called Lie types, or to one of twenty-six special cases, the so-called sporadic groups. Found his figure in. The complexity of this proof is astonishing, and a large number of mathematicians have made unremitting efforts. By the time it was published in 2004, the relevant literature had exceeded 100,000 pages.
In essence, simple groups are the basic building blocks of all finite groups, and their role is similar to that of prime numbers in natural numbers. However, a characteristic of simple groups is that these "building blocks" do not always uniquely identify a group, since there may be many different non-isomorphic groups that all have the same series of combinations. Daniel Gorenstein and his team are now working to simplify and revise this massive proof.
"The classification of finite simple groups is a unique achievement in mathematics, which has had a profound impact on many branches of mathematics."
Statement of the Classification Theorem
The classification theorem has practical value in many areas of mathematics, because when it comes to problems involving the structure of finite groups, the study can often be reduced to the problem of the properties of finite simple groups. Thanks to the derivation of this classification theorem, some related problems can even be solved by examining every simple group and every spontaneous group.
However, in the 1960s, Gorenstein announced in 1983 that the classification of finite simple groups had been completed, but this was premature due to a misunderstanding of some important evidence. The missing piece was not officially filled until 2004, with the publication of a 1,221-page proof by Aschbacher and Smith.
Overview of the Certification
The proof process can be broken down into several main parts. For example, in the classification of groups of small order 2, most of the groups are groups of small order Lie type, plus five alternating groups, seven characteristic type 2 groups, and nine spontaneous groups. In particular, when order 2 is 0, such groups are solvable, a result related to the Feit-Thompson theorem.
As for the classification of small 2nd-order groups, we need to consider a lot of situations: there are not only 26 spontaneous groups, but also 16 Lie-type groups, and many other peculiar behaviors of small-order groups, which must be considered in different Deal with each case one by one. According to the second-order decomposition of the group, it is necessary to divide it into an element type group and a characteristic type 2 group.
"This huge classification process is like a tough marathon for mathematics, and every detail needs to be carefully crafted."
History of Proof
In 1972, Gorenstein began a multi-year project to complete the classification of finite simple groups. The project consisted of 16 steps, focusing on the properties and structure of different types of groups. As the work progresses, the classification of most groups has been basically completed, but there is still a small number of groups that need more in-depth discussion and confirmation.
By 1985, the first generation of proofs had been completed, but because they were too cumbersome, the mathematical community began to revise the proof process. This so-called second-generation proof hopes to restate this huge theorem in a more concise and clear way. Most of the relevant members have rich experience and knowledge, which paves the way for new proofs.
Although progress has been slow, the project has already accrued to ten volumes and is expected to eventually reach five thousand pages. This length is partly due to the fact that the new proof uses a more relaxed style rather than the neat formalism on which the earlier proof was based.
In the end, this classification movement eventually became an important milestone in the mathematical community and provided a strong foundation for future mathematical development. So, what profound impact does this huge mathematical proof have on our understanding of mathematics?
