# Question about primitive roots and multiplicative groups

Let $$p$$ be an odd prime and let $$k \in \mathbb{N}$$. We know that $$U(\mathbb{Z}_p) = \{\overline{1},\overline{2},\dots,\overline{p-1}\}$$. Can it happen and when that $$U(\mathbb{Z}_p) = \{\overline{1}^k,\overline{2}^k,\dots,\overline{p-1}^k\}$$? Well it is obvious that when $$k=1$$ then this holds for every prime but my question is that can it and when does it happen when $$k \neq 1$$?

• It is a basic fact that this happens if and only if $k$ is co-prime with $p-1$; any standard text in elementary number theory contains this in some form. – W-t-P Apr 21 at 19:12