Permutation is Cyclic iff At Most One Non-Trivial Orbit

Theorem
Let $S$ be a set.

Let $\rho: S \to S$ be a permutation on $S$.

Then:
 * $\rho$ is a cyclic permutation


 * $S$ has no more than one orbit under $\rho$ with more than one element.
 * $S$ has no more than one orbit under $\rho$ with more than one element.

Also see
Some sources use this result as a definition for a cyclic permutation.