I'm reading this paper by Marcel Herzog on jstor: http://www.jstor.org/stable/2040939?seq=1
I want to follow up on a few things about the short proof of Theorem 1, found on the bottom of page 1 of the pdf.
The proof takes $E$ to be a elementary abelian $p$-group of maximal order, how can we be sure one exists?
I understand that each orbit divides the order of $E$, by the orbit-stabilizer theorem, so why does $i_p(G)\equiv f\pmod{p}$?
Here $i_p(G)$ is the number of elements of order $p$ in $G$, and $f$ is the number of fixed points on the set of elements of order $p$ when acted on by conjugation by $E$.