Corollary

Let $a$ and $b$ be two integers $> 0$. Then $v_p(ab) = v_p(a) + v_p(b)$. For $a$ to divide $b$, it is necessary and sufficient that $v_p(a) \leq v_p(b)$ for every prime number $p$.

In any group $G$, if the (monogenous) subgroup generated by an element $x \in G$ is of finite order $d$, $x$ is called an element of order $d$; the number $d$ is therefore the least integer $> 0$ such that $x^d = e$; if the subgroup generated by $x$ is infinite, $x$ is said to be of infinite order. These definitions, together with Proposition 4 (no. 4), imply in particular that in a finite group G the order of every element of G is a divisor of the order of G.