Definition:Characterization Theorem

Definition
Let $E$ be a non-abelian finite simple group.

Let $u \in E$ be a self-inverse element of $E$.

Let $H = \map {C_E} u$ be the centralizer of $u$ in $E$.

Let $G$ be a finite simple group with a self-inverse element $t$ such that $H \cong \map {C_G} t$.

A characterization theorem is a theorem that proves there is only one such group type $G$.

That is, that $G \cong E$ necessarily.