Groups of order 65

This article gives information about, and links to more details on, groups of order 65
See pages on algebraic structures of order 65 | See pages on groups of a particular order

Up to isomorphism, there is a unique group of order 65, namely cyclic group:Z65, which is also the external direct product of cyclic group:Z5 and cyclic group:Z13.

The fact of uniqueness follows from the classification of groups of order a product of two distinct primes. Since $15=5\cdot 13$ and $5$ does not divide $(13-1)$, the number $65$ falls in the one isomorphism class case.

Another way of viewing this is that $65$ is a cyclicity-forcing number, i.e., any group of order 65 is cyclic. See the classification of cyclicity-forcing numbers to see the necessary and sufficient condition for a natural number to be cyclicity-forcing.