If anyone can shed some light on the problem below (even if your unsure of the answer), your help would be much appreciated.
Let be a positive non-square integer and a prime number.
Consider the set
In this set, we define multiplication as follows:
The questions I'm interested in are the following:
1) Does this set contain primitive elements regardless of ? That is, does there always exist some element such that .
2) If the answer to the above is in the negative, then when is the multiplicative order of S less than ? When is it equal to ?