Prime Group is Cyclic

Theorem
Let $p$ be a positive prime integer.

Then there is only one group $G$ of order $p$, up to isomorphism, and it is the cyclic group of order $p$.

Each of its elements other than the identity is of order $p$, and therefore a generator of $G$.

Proof
The order of any element of a group divides the order of the group.

That is, if $x \in G$ then $\order x \divides p$.

From the definition of a prime number, the only positive integers that divide $p$ are $1$ and $p$.

So if $G$ has order $p \in \mathbb P$, then the order of any element is $1$ or $p$.

The only element of order $1$ is the identity.

Thus any $a \in G: a \ne e$ has order $p$ and therefore generates $G$.

By definition, $G$ is therefore cyclic.