8
$\begingroup$

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 $G$?

  • 1
    For one there is Euler's generalization of Fermat's little theorem. – 2017-01-21

2 Answers 2

3

I know one, although it isn't terribly exciting.

We can prove that $k!n!^k$ is a divisor of $(nk)!$ by noting that the first is the order of the subgroup of the permutations of $\{1,2,\dots kn\}$ such that it permutes the groups $\{1,2,3,\dots n\},\{n+1,\dots, 2n\} , \dots , \{kn-k+1,\dots,kn\}$ internally and externally.

  • 1
    This can also be proved by noting this is the number of ways to split $nk$ distinguishable students into $k$ indistinguishable groups of $n$ students each. – 2017-01-21
2

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 $n$.