Isomorphisms Between Multiplicative and Additive Modulo Groups.

Question

While working with the fundamental theorem of Abelian groups, I noticed something and would like to confirm my assumption.

$$(\mathbb{Z}/n\mathbb{Z})^* \cong \mathbb{Z}/φ(n)\mathbb{Z} $$

Is the above true? If so why exactly?

Answer 1

Your statement is true for some $n$, but not for all $n$. For example, it's not true for $n=8$.

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Addendum: I recommend this Wikipedia article, which states that

the group $ (\mathbb{Z}/n\mathbb{Z})^\times$ is cyclic if and only if $n$ is $ 1, 2, 4,$ $p^k$ or $2p^k$,

where $ p$ is an odd prime and $k > 0$.