Talk:Centralizer of Self-Inverse Element of Non-Abelian Finite Simple Group is not That Group

In : $1$: Introduction to Finite Group Theory: $1.14$, the result is posed as:


 * (R. Brauer and K. A. Fowler [a8], 1955) Let $G$ be a non-abelian finite simple group (so that, by $1.12$, $\order G$ is even) and let $t$ be an involution in $G$. Then $\map {C_G} t \ne G$, and if $\order {\map {C_G} t} = m$ then $\order G \le \paren {\frac 1 2 m \paren {m + 1} }!$

where reference [a8] is the one cited.

In $1.13$ he states:


 * Let $G$ be any group of even order. Then $G$ possesses at least one element of order $2$. (Any such element is called an involution.)

This gives the definition of involution.

I have not inspected Brauer and Fowler to see what it says, but that looks like the next stop along the way. Feel free to take this up. --prime mover (talk) 10:07, 20 May 2020 (EDT)