# 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.

## Historical Note

If a **characterization theorem** does not exist for a given non-abelian finite simple group $E$, that means there may be a previously unknown simple group among the groups $G$ predicted by Upper Bound of Order of Non-Abelian Finite Simple Group.

Hence proving the non-existence of a **characterization theorem** such a group $E$ was a powerful technique used in the Classification of Finite Simple Groups, mainly during the $1970$s.

## Sources

- 1978: John S. Rose:
*A Course on Group Theory*... (previous) ... (next): $1$: Introduction to Finite Group Theory: $1.15$