= Prime-power characterization by full permutation cycles
{title2=$\forall\sigma\in S_n:\ \operatorname{ord}(\sigma)=n\Longrightarrow\sigma\text{ is an }n\text{-cycle}\quad\iff\quad n=p^r$}
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 $a+b\leq ab=n$, disjoint cycles of lengths $a$ and $b$ fit inside $S_n$. Their product has <order of a permutation> $\operatorname{lcm}(a,b)=n$, but is not a full cycle.
Back to article page
