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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 $k \neq 1$?

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    $\begingroup$ 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. $\endgroup$ – W-t-P Apr 21 at 19:12

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