Let $p$ be a prime of the from $p= 2^k +1$ where $k$ is natural number.
The first question that was asked is to prove that the set of all generators of $\Bbb Z^*_p$ is not a subgroup of $\Bbb Z^*_p$.
By brute force and Euler's theorem, I saw that the possible sizes of the subgroup are $1$, $2$ and $4$ for $k =2$ and $p=5$.
Then I saw that $1$ always would not be a generator of any group, and because of that, the set of all the generators will not include $1$, which means that the identity element will not be included and therefore it could not be a subgroup.
I was trying to show that the set of all elements of $\Bbb Z^*_p$ that are not generators is a subgroup of $\Bbb Z^*_p$.
I can easily see that in $\Bbb Z^*_5$ it's true. I started checking it for $\Bbb Z^*_{17}$, but it's a lot of work. I saw that $2$ is not a generator and $3$ is a generator. It's a lot of work.
Is there any way to quickly check the list of the generators of $\Bbb Z^*_{17}$ ? How do I prove it?