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Let $\sigma$ be a non-identity permutation of $S_n$, can we find an element $\rho$ in $S_n$ so that $\{\sigma\}\cup\{\rho\}$ are generators of $S_n$? It is known that $S_n$ is generated by $(12)$ and $(123\dots n)$, and the probability that two random permutations generate $S_n$ approaches $\frac{3}{4}$ as $n$ increases by Dixon's theorem, but I'm not sure if this is true.

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    $\begingroup$ This is true except when $n = 4$. See the first theorem of fmf.uni-lj.si/~potocnik/poucevanje/SeminarI2013/… $\endgroup$ – KCd Jan 3 '15 at 3:54
  • $\begingroup$ I meant that if you copy paste your comment into an answer I would gladly upvote it and accept it, this is great. $\endgroup$ – HereToRelax Jan 3 '15 at 4:01
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Since KCd didn't put it into an answer, I will (just to get it out of the unanswered queue):

This is true for all symmetric groups except $S_4$. See Isaacs and Zieschang, Theorem A.

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