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


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.



Sources