Finite Simple Groups

1899 年发表了第一篇分类无限系列单群的论文. ("Group theory is an important branch of mathematics, and it has important applications in mathematics, physics, chemistry and other fields. The classification theorem for the finite simple groups, which was completed in 2004, is one of the most important mathematical achievements of the 20th century. In terms of the number of participating researchers, the number of published papers, and the total length of the proof (more than 15,000 pages), it is unprecedented in the history of mathematics. The first paper classifying an infinite family of finite simple groups, starting from a hypothesis on the structure of certain proper subgroups, was published by Burnside in 1899.") 近年来, 群表示论中 McKay 猜想和 Alperin 猜想等重要课题已经约化为单群和拟单群情形 (参 见文献 [146]), 迫切地需要理解和识别单群更多的性质. 著名群论专家 Solomon [186] 指出: "目前正在 进行的项目是出版一个超过 12 卷的系列书, 以展示有限单群分类定理的完整证明, 预计该项目将于 2025 年完成." ("The ongoing project to publish a series of more than 12 volumes presenting a complete proof of this theorem is expected to be completed by 2025.") (上述两段英文摘自 R. Solomon 于 2020 年 11 月 2 日寄给作者的电子邮件, 也可参见文献 ). 作者于 1987 年向 Thompson 教授报告了上述猜想, 得到了 Thompson 的鼓励和充分肯定. Thompson 在回信中指出: "Good luck with your conjecture about simple groups. I hope you continue to work on it", "I like your arguments", "This would certainly be a nice theorem". 与此同时, Thompson 1) 2) 在 1987 和 1988 年的两封信中分别提出了如下的问题和猜想 (参见文献 [100, 问题 12.37-12.39]). 定义 Thompson 在他的信中指出: "The problem arose initially in the study of algebraic number fields, and is of considerable interest."

Many questions in additive number theory (Goldbach's conjecture, Fermat's Last Theorem, the Twin Primes conjecture) can be expressed in the language of sum and difference sets. As a typical pair contributes one sum and two differences, we expect |A−A|>|A+A|for finite sets A. However, Martin and O'Bryant showed a positive proportion of subsets of {0,…,n} are sum-dominant. We generalize previous work and study sums and differences of pairs of correlated sets (A,B)(a∈{0,…,n} is in A with probability p, and a goes in B with probability_(ρ1) if a∈A and probability_(ρ2)if a∉A). If |A+B|>|(A−B)∪(B−A)|, we call (A,B) a sum-dominant (p,_(ρ1,ρ2))-pair. We prove for any fixed ρ→=(p,ρ1,ρ2) in (0,1)^3, (A,B) is a sum-dominant (p,_(ρ1,ρ2))-pair with positive probability, which approaches a limit P(ρ→). We investigate p decaying with n, generalizing results of Hegarty–Miller on phase transitions, and find the smallest sizes of MSTD pairs

Many fundamental questions in additive number theory (such as Goldbach's conjecture, Fermat's last theorem, and the Twin Primes conjecture) can be expressed in the language of sum and difference sets. As a typical pair of elements contributes one sum and two differences, we expect that |A -A| > |A + A| for a finite set A. However, in 2006 Martin and O'Bryant showed that a positive proportion of subsets of {0, . . . , n} are sum-dominant, and Zhao later showed that this proportion converges to a positive limit as n → ∞. Related problems, such as constructing explicit families of sum-dominant sets, computing the value of the limiting proportion, and investigating the behavior as the probability of including a given element in A to go to zero, have been analyzed extensively. We consider many of these problems in a more general setting. Instead of just one set A, we study sums and differences of pairs of correlated sets (A, B). Specifically, we place each element a ∈ {0, . . . , n} in A with probability p, while a goes in B with probability We prove that for any fixed ρ = (p, ρ 1 , ρ 2 ) in (0, 1) 3 , (A, B) is a sum-dominant (p, ρ 1 , ρ 2 )-pair with positive probability, and show that this probability approaches a limit P ( ρ). Furthermore, we show that the limit function P ( ρ) is continuous. We also investigate what happens as p decays with n, generalizing results of Hegarty-Miller on phase transitions. Finally, we find the smallest sizes of MSTD pairs.

In earlier work it was shown that each nonabelian finite simple group G has a conjugacy class C such that, whenever 1 = x ∈ G, the probability is greater than 1/10 that G = x, y for a random y ∈ C. Much stronger asymptotic results were also proved. Here we show that, allowing equality, the bound 1/10 can be replaced by 13/42; and, excluding an explicitly listed set of simple groups, the bound 2/3 holds. We use these results to show that any nonabelian finite simple group G has a conjugacy class C such that, if x 1 , x 2 are nontrivial elements of G, then there exists y ∈ C such that G = x 1 , y = x 2 , y . Similarly, aside from one infinite family and a small, explicit finite set of simple groups, G has a conjugacy class C such that, if x 1 , x 2 , x 3 are nontrivial elements of G, then there exists y ∈ C such that G = x 1 , y = x 2 , y = x 3 , y . We also prove analogous but weaker results for almost simple groups.

We prove surjectivity of certain word maps on finite non-abelian simple groups. More precisely, we prove the following: if N is a product of two prime powers, then the word map (x, y) → x N y N is surjective on every finite non-abelian simple group; if N is an odd integer, then the word map (x, y, z) → x N y N z N is surjective on every finite quasisimple group. These generalize classical theorems of Burnside and Feit-Thompson. We also prove asymptotic results about the surjectivity of the word map (x, y) → x N y N that depend on the number of prime factors of the integer N . GURALNICK, LIEBECK, O'BRIEN, SHALEV, AND TIEP 5.3. Induction step: Symplectic groups 33 5.4. Induction step: Orthogonal groups 42 5.5. Completion of the proof of Theorem 1 for classical groups 48 6. Theorem 1 for exceptional groups 48 7. Odd power word maps 53 7.1. Preliminaries 53 7.2. Regular 2-elements in classical groups in odd characteristic 54 7.3. Proof of Theorem 2 for classical groups in odd characteristics 57 7.4. Proof of Theorem 2 for exceptional groups in odd characteristics 65 8. Asymptotic surjectivity: Proofs of Theorems 3 and 4 69 References 75

We prove the following three closely related results: (1) Every finite simple group G has a profinite presentation with 2 generators and at most 18 relations. (2) If G is a finite simple group, F a field and M is an F G-module, then dim H 2 .G; M / Ä .17:5/ dim M . (3) If G is a finite group, F a field and M is an irreducible faithful F G-module, then dim H 2 .G; M / Ä .18:5/ dim M .