Groups of order 15

Up to isomorphism, there is a unique group of order 15, namely cyclic group:Z15, which is also the external direct product of cyclic group:Z3 and cyclic group:Z5.

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.