A generalization of the Brauer-Fowler theorem (Q6634495)

The Brauer-Fowler theorem, which is one of the starting points of the research going to the classification of finite simple groups, says the following: \N\NTheorem. [\textit{R. Brauer} and \textit{K. A. Fowler}, Ann. Math. (2) 62, 565--583 (1955; Zbl 0067.01004)] There is a function \(f: \mathbb{N } \longrightarrow \mathbb{N}\) such that if \(G\) is a finite simple group and \(\iota\) is an element of order \(2\), then the order of \(G\) is less than \(f(|C_{G}(\iota)|)\). Clearly, this implies that for a given order of a centralizer of an involution, there are finitely many finite simple groups with such a centralizer.\N\NThe author, using the classification of finite simple groups, proves one step further. Namely, he proves that there is a function \(\psi: \mathbb{N } \longrightarrow \mathbb{N}\) such that if \(G\) is a finite simple group with an involution \(\iota\) such that \(C_{G}(\iota)\) contains \(k\) many involutions, then the order of \(G\) is less than \(\psi(k)\).