On the uniqueness of the finite simple groups with a given centralizer of a 2-central involution

Let $H$ be a finite group having center $Z(H)$ of even order. By the classical Brauer-Fowler theorem there can be only finitely many non isomorphic simple groups $G$ which contain a $2$-central involution $t$ for which $C_G(t) \cong H$. In this article we give several conditions which together suffice to prove that up to isomorphism there is a unique simple group $G$ having a $2$-central involution $z$ with centralizer $C_G(z) \cong H$. Together they yield a practical uniqueness criterion (Theorem 2.1). This is demonstrated by giving a new short uniqueness proof for Janko's sporadic simple group $J_1$. Similar, uniform uniqueness proofs are outlined for ten other sporadic simple groups.