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When Any Group of N Elements is Cyclic?

We give a simple proof of the well-known fact: any group of n elements is cyclic if and only if n and ϕ(n) are coprime. This note is accessible for students familiar with permutations and basic number theory. No knowledge of abstract group theory is required; a few necessary notions are introduced in the course of the proof. The note could also be an interesting easy reading for mature mathematicians.