I know the following result is true, but I haven't managed to find a proof to it:
A cyclic number is a positive integer such that $\phi(n)$ is coprime to $n$, where $\phi(n)$ denotes Euler's totient function. Let $G$ be a group of finite order $n$, so that $n$ is a cyclic number, using only the above definition for a cyclic number - then $G$ is a cyclic group.
This theorem has elegant (in my opinion) immediate results, like instantly showing that every group of prime order is cyclic, and also showing that up to isomorphism, there is only one group of order 15 - because 15 is a cyclic number, without using Sylow's theorems.
I can't see the connection between the way Euler's function acts on $n$ and the groups of its order. Would anyone like to shed some light, hopefully even providing a proof? Thanks a lot!