Centralizer of Self-Inverse Element of Non-Abelian Finite Simple Group is not That Group

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

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

Then:
 * $\map {C_G} t \ne G$

where $\map {C_G} t$ denotes the centralizer of $t$ in $G$.

Proof
By the Feit-Thompson Theorem, $G$ is of even order.