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 \phi(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. 
V. Bragin, Ant. Klyachko and A. Skopenkov
math.GR math.HO