"Order" in cryptographic terms for generators Frequently I have seen people use the term order in cryptography (the group theoretic one). I have a mathematical background and order (say for prime modulus $p$) is defined as the smallest integer such that 

$$a^r \equiv 1 \pmod p$$

So a generator (something which has the max possible order i.e order is $\phi(p)=p-1$) will have order (p-1). In case of a composite group, the generator should have an order $=\phi(N)$.

How is it that frequently I read in Crypto literature that people say generator has order N?

For example: The first answer at: When do we need composite order groups for bilinear maps and when prime order?

Am I missing some details or is my understanding incorrect?
You probably know that cyclic groups of any order exist, so it is perfectly possible to have a cyclic group of order $pq$, and to consider a generator thereof.