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As the title says, I would like to know if there is an interesting form to express (isomorphically) $Aut(\mathbb{Z}_n)$ as a direct product $Z_{m_1}\oplus \ldots \oplus Z_{m_n}$ for some integers $m_1,\ldots$, $m_n$. This question arise from reading Hungerford T.W. Algebra. (Page 34) The way he ask for a solution is a little ambiguous. I already know that $Aut(\mathbb{Z}_n)$ is isomorphic to $U(\mathbb{Z}_n)$.

Julio
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  • At least we know that order of $Aut(Z_n)$ is $\phi(n)$ where it is euler phi function. And When $n$ is prime it is isomorphic to $Z_{n-1}$ As far as I know there is no general clasification except some special case. – mesel Feb 24 '14 at 22:45
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    Use the Chinese remainder theorem to reduce to the case that $n$ is a prime power and then look up the classical results on existence of primitive roots. – Qiaochu Yuan Feb 25 '14 at 01:27

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