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.
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Sources
- 1955: Richard Brauer and K.A. Fowler: On groups of even order (Ann. Math. Ser. 2 Vol. 62: 565 – 583) www.jstor.org/stable/1970080
- 1978: John S. Rose: A Course on Group Theory ... (previous) ... (next): $1$: Introduction to Finite Group Theory: $1.14$
