The primary purpose of our note is to prove Theorem. The background results from additive number theory are given in Section 2. These concern the structure of sets with small doubling. The technical aspects of the proof are given in Section 3, roughly as follows. On the one hand, the statement of the theorem says something about the possible orders of a basis for Zn when that order is large, namely of order n. On the other hand, various results from additive number theory imply that if A is a basis for Zn, then the iterated sumsets hA cannot grow in size ‘too slowly’ and, if the growth rate is close to the slowest possible, then A has a very restricted structure. Putting these two things together allows us to describe closely the structure of (a small multiple of) a basis A of large order, and from there we can establish the result...