Referring to the group of units.

My first thought is yes, since $1$ would be the generator. 

Although I think I'm getting confused between the generator and the identity element in this case. $1$ is definitely the latter, is it also a generator?

I believe I'm doing the operation incorrectly. For example, $3 \cdot 3 =1$ correct? $3$ couldn't be the generator because if I kept doing this operation I would start repeating the same elements, but not all the elements ($1,3,5,7)$