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


Sources