Brauer-Fowler Bounds for Elements of Odd Prime Order

Abstract

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 \(e\).