Let $G$ be a group whose order is equal to $p$, a prime number. Then $G$ is isomorphic to the cyclic group $C_p$ of order $p$.

Proof

Pick any non-identity element $g$ of the group.

By Lagrange's theorem, the subgroup generated by $g$ has size $1$ or $p$ (since $p$ was prime). But it can't be $1$ because the only subgroup of size $1$ is the trivial subgroup.

Hence the subgroup must be the entire group.