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

?