# Groups of order 15

The fact of uniqueness follows from the classification of groups of order a product of two distinct primes. Since $15 = 3 \cdot 5$ and $3$ does not divide $(5 - 1)$, the number $15$ falls in the one isomorphism class case.
Another way of viewing this is that $15$ is a cyclicity-forcing number, i.e., any group of order 15 is cyclic. See the classification of cyclicity-forcing numbers to see the necessary and sufficient condition for a natural number to be cyclicity-forcing.