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 $\map {C_G} t \ne G$.
So trivial it's scary: something stronger was proved
Remark: the identity is a self-inverse element with $\map {C_G} e = G$ trivially
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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: $1.14$
