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.

If $$x \in G$$ then $$\left|{x}\right| \backslash p$$.

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.