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 definition of a simple group 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:

$\map Z G = G$ if and only if $G$ is abelian

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

Thus $t \notin \map Z G$.

From definition of center:

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

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

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

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: $1.14$