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

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## Corollary to Upper Bound of Order of Non-Abelian Finite Simple Group

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.

$\blacksquare$

## Sources

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