Groups of order 15
This article gives information about, and links to more details on, groups of order 15
See pages on algebraic structures of order 15| See pages on groups of a particular order
The fact of uniqueness follows from the classification of groups of order a product of two distinct primes. Since and does not divide , the number falls in the one isomorphism class case.
Another way of viewing this is that 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.