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

Let $t \in G$ which is not the identity.

By and Center of Group is Normal Subgroup:
 * either $\map Z G = G$ or $\map Z G$ is the trivial group.

By definition of an abelian group:
 * $\map Z G = G$ $G$ is abelian

Hence we must have $\map Z G$ is the trivial group.

Thus $t \notin \map Z G$.

From :
 * $\exists x \in G: t x \ne x t$

For this $x$, $x \notin \map {C_G} t$.

Hence $\map {C_G} t \ne G$.