On the uniqueness of the finite simple groups with a given centralizer of a \(2\)-central involution (Q1409622)

Let \(H\) be a finite group such that its center has even order. The Brauer-Fowler theorem states that there are only finitely many isomorphism types of finite simple groups \(G\) such that \(G\) has a 2-central involution (an involution in the center of some Sylow 2-subgroup) with \(C_G(t)\simeq H\). In a predecessor of the paper under review [J. Algebra 234, No. 2, 668-693 (2000; Zbl 0971.20009)] the author showed that under suitable assumptions there is a deterministic algorithm for the construction of all finite simple groups with the above property. In this paper the author provides a sufficient criterion which ensures that up to isomorphism there is only one simple group \(G\) with \(C_G(t)\simeq H\). The author uses this criterion for a short uniqueness proof of the sporadic simple Janko group \(J_1\). He also outlines similar uniqueness proofs for 10 other sporadic simple groups.