TITLE: If $p\mid|G|$ then how many elements of order $p$ are there in $G$?
QUESTION [5 upvotes]: Let $G$ be a finite group and $p$ be a prime such that $p\mid|G|$ , then obviously $G$ has an element of order $p$ (by Cauchy's theorem) ; I would like to know exactly how many elements of order $p$ are there in the group ? Please help

REPLY [3 votes]: Something more can be said. Let's stick to the notation of Andrea Mori - let $M(p)$ denote the number of elements of order $p$ of $G$, then $$M(p) \equiv -1 \text { mod } p.$$
This follows easily from Sylow Theory or from the proof of James McKay of Cauchy's Theorem on the existence of elements of order $p$, if $p$ divides $|G|$.