Groups of prime order are cyclic

Statement

Let $p$ be a prime number and let $G$ be a group of order $p$. Then $G$ is cyclic, and each element of $G$ other than the identity generates $G$.

Remarks

If $g \neq e$, then the order of $\langle g \rangle$ is greater than $1$ and divides $p$ by Lagrange's theorem, so it is $p$. Thus each group of order $p$ is isomorphic to $\mathbb{Z}/p\mathbb{Z}$. A group of prime order has no subgroups except $\{e\}$ and itself.

Explanations

L1 human

When the size of a group is a prime, there is no room for proper subgroups other than the trivial one. Any element other than the identity generates a subgroup larger than $1$, so it generates everything. Thus, for example, each group with $7$ elements is a clock with $7$ hours.