TITLE: Lagrange's theorem and divisibility consequences.
QUESTION [8 upvotes]: There are some simple, but sometimes intriguing, divisibility statements
that are straightforward consequences of Lagrange's theorem. For instance:

$p$ divides $a^{p-1}-1$ (Fermat's little thm)
$n!$ divides $(p^n-1)(p^n-p)\cdots(p^{n}-p^{n-1}).$

The latter one can be derived  from the fact that $S_n  \hookrightarrow GL_{n}(\mathbb{F}_p)$.
I've noticed  that  simple   examples like those  can be very compelling  for students (begginers).
Question: Are there more interesting divisibility statements that are immediate conseguences of Lagranges' thm? That is, coming from the simple fact a group $H$ is a subgroup of a finite group $G$?

REPLY [2 votes]: For $a,n \geq 1$, $n\mid\phi(a^n - 1)$ ($\phi$ being Euler's $\phi$-function). This follows from noting that the order of $a$ in $\left(\Bbb Z/(a^n-1)\Bbb Z\right)^\times$ is $n$.