Wikipedia says:
It is known that $(\mathbb{Z}/n\mathbb{Z})^\times$ is cyclic if and only if n is 1 or 2 or 4 or $p^k$ or $2p^k$ for an odd prime number p and k ≥ 1.
The statement seems provable given the definition and properties of the Carmichael function.
I need a reference (preferably a book) to read more on cyclic group, their order, and the Carmichael function. I tried several book (on number theory or algebra), but I couldn't find one that stated and proved the above theorem.
Could you please recommend one?