Prime Group is Cyclic

Theorem
Let $p$ be a prime number.

Let $G$ be a group whose order is $p$.

Then $G$ is cyclic.

Proof
Let $a \in G: a \ne e$ where $e$ is the identity of $G$.

From Group of Prime Order p has p-1 Elements of Order p, $a$ has order $p$.

Hence by definition, $a$ generates $G$.

Hence also by definition, $G$ is cyclic.