奇素数阶元素的Brauer-Fowler界
Brauer-Fowler Bounds for Elements of Odd Prime Order
AI总结：
本文研究奇素数阶元素的Brauer-Fowler界，证明Skresanov的对合相关结果的直接类似物对奇素数阶元素不成立，给出了有限单群阶数由中心化子内p阶元素数量及指数界定的结论。
AI中文摘要：
Brauer-Fowler定理通过对合的中心化子的阶数来界定有限单群的阶数；Hartley证明了其自同构版本，通过自同构的阶数及其不动点子群的阶数来界定有限单群的阶数。Strunkov提出，在对合情形下，中心化子的阶数能否替换为与给定对合可交换的对合的数量，这一问题最近被Skresanov肯定地解答。受该问题启发，与Hartley使用不动点子群的阶数不同，我们研究从这类子群内指定素数阶元素的计数中能推导出什么。我们表明，Skresanov结果的直接类似物对奇素数阶元素不成立；我们证明，若G是有限单群，x∈G有奇素数阶p，C_G(x)包含至多k个p阶元素，且C_G(x)的指数至多为e，则|G|可由k和e界定。
英文摘要：
The Brauer--Fowler theorem bounds the order of a finite simple group in terms of the order of the centralizer of an involution. Hartley proved an automorphism version, bounding the order of a finite simple group in terms of the order of an automorphism and the order of its fixed-point subgroup. Strunkov asked whether, in the involution case, the order of the centralizer can be replaced by the number of involutions commuting with the given involution; this was recently answered affirmatively by Skresanov. Motivated by this question, and in contrast with Hartley's use of the order of a fixed-point subgroup, we study what can be deduced from counting elements of prescribed prime order inside such a subgroup. We show that the direct analogue of Skresanov's result fails for elements of odd prime order. We prove that if \(G\) is a finite simple group, \(x\in G\) has odd prime order \(p\), \(C_G(x)\) contains at most \(k\) elements of order \(p\), and the exponent of \(C_G(x)\) is at most \(e\), then \(|G|\) is bounded in terms of \(k\) and \(e\).
