# 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

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: $1.14$