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I am new to group theory, and I observed that some multiplicative mod-n groups are cyclic and some are not . For ex :

$U(8) = (1,3,5,7)$ is not cyclic , but

$U(10) = (1,3,7,9)$ is cyclic and its generators are $(1,3,9)$.

How do we find which multiplicative group is cyclic, which is not?

Bernard
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Vinay Varahabhotla
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  • Hint: Using the Chinese remainder theorem you can reduce your problem to the case that $n$ is a prime power (for $m,n$ coprime $(Z/(mn))^\times = (Z/m)^\times\times(Z/n)^\times$)) and $(Z/p^k)^\times$ is cyclic unless $p=2$ and $k\ge 2$. – j.p. Jan 06 '19 at 18:03
  • @j.p. Right, this is exactly how the answer (at the duplicate) starts:) – Dietrich Burde Jan 06 '19 at 19:24

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