Group of Order 15 is Cyclic Group

Theorem
Let $G$ be a group whose order is $15$.

Then $G$ is cyclic.

Proof
We have that $15 = 3 \times 5$.

Thus:
 * $15$ is square-free
 * $5 \equiv 2 \pmod 3$
 * $3 \equiv 3 \pmod 5$

The conditions are fulfilled for all groups of order $15$ to be cyclic.