PDF version \begin{theorem} If $g$ is an element of $G=S_n$ then $C_G(g)=\langle g\rangle$ if and only if the cycles of $g$ are of unequal coprime lengths.\\ Alternatively, $C_G(g)=\langle g\rangle$ if and only if $g$ has cycles of coprime length with at most one 1-cycle. \end{theorem} \begin{proof} Since powers of $g$ centralise $g$ it follows that $\langle g\rangle\subseteq C_G(g)$ so \begin{align} C_G(g)=\langle g\rangle\iff \lvert\langle g\rangle\rvert=\lvert C_G(g)\rvert\iff\lvert g\rvert=\lvert C_G(g)\rvert \end{align} (1) Let $G$ act on itself by conjugation. Then $xgx^{-1} = x\iff xg=gx$ so \begin{equation} \Stab(g)=C_G(g) \end{equation} (2) \begin{align*} \lvert\Orb(g)\rvert&=\text{\small the number of distinct conjugates of g}\\ &=\text{\small the number of permutations with the same cycle structure as }g \end{align*} Let $a$ be the product of the lengths of the cycles of $g$\\ Let $b$ be the number of cycles of equal length.\\ Let $l$ be the least common multiple of the lengths of the cycles, $\lvert g\rvert=l\leq a$.\\\\ Then the number of permutations with the same cycle structure as $g$ is \begin{align*} \frac{n!}{ab}=\frac{\lvert G\rvert}{ab} \end{align*} Hence \begin{equation} \lvert\Orb(g)\rvert=\frac{\lvert G\rvert}{ab} \end{equation} (3) By the Orbit-Stabiliser theorem, $\lvert\Orb(g)\rvert\times\lvert\Stab(g)\rvert=\lvert G\rvert$ so by (3) \begin{align*} \lvert\Stab(g)\rvert=\frac{\lvert G\rvert}{\lvert\Orb(g)\rvert}=\frac{\lvert G\rvert}{\lvert G\rvert/ab}=ab\geq\lvert g\rvert b\geq\lvert g\rvert=l \end{align*} and it follows that \begin{align} \lvert\Stab(g)\rvert=\lvert g \rvert\iff a = l \text{ and } b = 1 \end{align} (4) But \begin{align} a=l\iff g\text{ has coprime cycle lengths}\\ b=1 \iff g \text{ has unequal cycle lengths} \end{align} (5) (6) The result now follows from (1), (2), (4), (5) and (6). \[ C_G(g)=\langle g\rangle\iff\lvert\Stab(g)\rvert=\lvert g \rvert\iff\text{the cycles of $g$ are of unequal coprime lengths} \] Cycles of coprime length will have unequal lengths unless they are 1-cycles. Hence we may replace \textit{unequal} by \textit{at most one 1-cycle}. \end{proof}