Upper Bound of Order of Non-Abelian Finite Simple Group/Corollary

Theorem
Let $H$ be a finite group of even order.

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

Then there are finitely many types of finite simple group $G$ such that:
 * $G$ has a self-inverse element $t \in G$
 * $\map {C_G} t \cong H$

Proof
First suppose that $G$ is abelian.

Then by Abelian Group is Simple iff Prime, $\order G = 2$.

So let $G$ be non-abelian.

From Upper Bound of Order of Non-Abelian Finite Simple Group:


 * $\order G \le \paren {\dfrac {\order H \paren {\order H + 1} } 2}!$

which depends completely upon the given group $H$.

The result follows from Finite Number of Groups of Given Finite Order.