Prime-power characterization by full permutation cycles ($\forall \sigma \in S_n:\mathrm{ord}(\sigma )=n⟹\sigma \text{ is an }n\text{-cycle}⟺n=p^r$)

For $n>1$, every permutation of order of a permutation $n$ in $S_n$ is one full cycle if and only if $n$ is a prime power. If $n=p^r$, all lengths in its disjoint cycle decomposition divide $p^r$, and their least common multiple is their maximum. One cycle therefore has length $n$ and uses every symbol. If $n$ has more than one prime divisor, write $n=ab$ with $a,b>1$ coprime by separating one full prime-power factor from the others. Since $a+b\le ab=n$, disjoint cycles of lengths $a$ and $b$ fit inside $S_n$. Their product has order of a permutation $\mathrm{lcm}(a,b)=n$, but is not a full cycle.