Wilson's theorem: for every prime $p$, $(p-1)!\equiv-1\pmod p$

Given: A prime $p$.

The unit classes modulo $p$ other than $[1]_p$ and $[-1]_p$ occur in disjoint inverse pairs, while those displayed classes are the only self-inverse ones; at $p=2$ they coincide (In $(\mathbb{Z}/p)^\times$, inversion pairs every class except $[1]_p$ and $[-1]_p$, which are the only self-inverse classes).

The nonzero standard representatives modulo $p$ are $1,\ldots,p-1$, and their product is $(p-1)!$ (For $n\ge 1$, every class in $\mathbb{Z}/n$ has one representative $r$ with $0\le r<n$, so $\lvert\mathbb{Z}/n\rvert=n$; while $\mathbb{Z}/0$ is in bijection with $\mathbb{Z}$, The factorial $n!$ and the falling factorial $n^{\underline{k}}$, defined by recursion in $\mathbb{N}$).

A finite product in a commutative monoid may be regrouped and reordered, and each inverse pair has product $1$ (The product $g_0 g_1 \cdots g_{n-1}$ of a finite list in a monoid, by recursion, with the empty product ($n = 0$) equal to the identity, Generalised associativity: in a monoid the product of a finite list does not depend on the bracketing, and in a commutative monoid it does not depend on the order of the factors either, For every natural $n$, $(\mathbb{Z}/n,+)$ is an abelian group, multiplication is a commutative monoid operation, and both distributive laws hold).

Equality of residue classes modulo $p$ is equivalent to congruence modulo $p$ (The congruence class $[a]_n$ and the quotient set $\mathbb{Z}/n$), and congruence means divisibility of the difference (Congruence modulo an integer: $a\equiv b\pmod n$ when $n\mid(a-b)$, including the moduli $0$ and $1$).

The quotient $\mathbb Z/p$ is a field, so every nonzero class is a unit (For every prime $p$, the two operations on $\mathbb{Z}/p$ make it a field).

Products of residue classes are computed by multiplying representatives: $[a]_p[b]_p=[ab]_p$ (Addition and multiplication on $\mathbb{Z}/n$ by $[a]_n+[b]_n=[a+b]_n$ and $[a]_n[b]_n=[ab]_n$).