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

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## Theorem

Let $G$ be a non-abelian finite simple group.

Let $t \in G$ be a self-inverse element of $G$.

Let $\map {C_G} t$ denote the centralizer of $t$ in $G$.

Let $m = \order {\map {C_G} t}$ be the order of $\map {C_G} t$.

Then:

- $\order G \le \paren {\dfrac {m \paren {m + 1} } 2}!$

### Corollary

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

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## Sources

- 1955: Richard Brauer and K.A. Fowler:
*On groups of even order*(*Ann. Math.***Ser. 2****Vol. 62**: pp. 565 – 583) www.jstor.org/stable/1970080

- 1978: John S. Rose:
*A Course on Group Theory*... (previous) ... (next): $1$: Introduction to Finite Group Theory: $1.14$