The group of units of Z/pZ is cyclic

Statement

Let $p$ be a prime number. Then the group $(\mathbb{Z}/p\mathbb{Z})^\times$ is cyclic of order $p - 1$.

Remarks

So $(\mathbb{Z}/p\mathbb{Z})^\times$ is isomorphic to the additive group $\mathbb{Z}/(p-1)\mathbb{Z}$, but no simple general formula for a generator is known. A generator is a primitive root modulo $p$, and there are $\varphi(p - 1)$ of them. For $p = 7$, the powers of $3$ modulo $7$ are $1, 3, 2, 6, 4, 5$, so $3$ is a primitive root modulo $7$. More generally, $(\mathbb{Z}/n\mathbb{Z})^\times$ is cyclic if and only if $n$ is $1$, $2$, $4$, $p^k$ or $2 p^k$ for an odd prime $p$. Gauss first proved this.

Explanations

L1 human

For a prime $p$, one residue modulo $p$ has powers that give all the nonzero residues. Such a residue is a primitive root. So the multiplicative group modulo $p$ is cyclic.