Fermat's Little Theorem/Proof 5

Theorem

Let $p$ be a prime number.

Let $n \in \Z_{>0}$ be a positive integer such that $p$ is not a divisor of $n$.

Then:

$n^{p - 1} \equiv 1 \pmod p$

Proof

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Let $S$ be a set of n elements, and consider $p$-tuples $\left( a_1, a_2, \dots, a_p \right)$ and consider the Group Action of these p-tuples by $\Z/p\Z$ via cyclic shifts.

For example, if:

$S = \{ b, r, g \}$

and:

$p=5$

then $rgbgg$ is equivalent to:

$grgbg, ggrgb, bggrg, gbggr$.

We use Burnside's Lemma, counting the number of elements fixed by each element of the group $\Z/p\Z$. Note that the identity fixes all $a^p$ elements, and the only tuples fixed by any of the other $p-1$ elements of $\Z/p\Z$ are the $a$ tuples that are made up of only one element of $S$, for example $rrrrr$, $ggggg$, and $bbbbb$.

Burnside's Lemma says that:

$p \mid \left( n^p + (p-1) n \right)$

equivalently:

$n^p \equiv n \pmod p$

$\blacksquare$