\begin{definition}[Definition:Characterization Theorem]
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.
\end{definition}