Let $G$ be a finite simple group with $P\in{\rm Syl}_2(G)$ being an elementary abelian group. Suppose $ P=C_G(x) $ for all $x\in P\setminus \{1\}$. Show that every element in $G$ is either an involution or of odd order.
I got stuck on this question for hours. I do not know how to put all these conditions together. For example, I have no idea how to use the simplicity and the structure of the Sylow $2$-subgroups here. I have tried to consider group actions but no luck. Any hint is appreciated! Thanks.
Take an element $x$ from $G$ of even order but not an involution. Then $y=x^k$ is of order $2$ for some $k$. Then by your condition the centralizer of $y$ in $G$ consists of involutions. This is a contradiction because $x$ is in the centralizer of $y$.
The simplicity of $G$ is not needed.