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If a permutation $\sigma$ $\in$ $S_n$, the permutation group of n elements, and $\sigma$ can be expressed as a product of disjoint cycles, is it necessary that the disjoint cycles be elements in $\langle{\sigma}\rangle$, the cyclic subgroup of $S_n$ generated by $\sigma$?

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  • $\begingroup$ What do you mean by "expressed uniquely"? Disjoint cycles commute. $\endgroup$ Jul 15, 2015 at 11:51
  • $\begingroup$ You're correct- I was thinking of something else while writing it. Good catch. $\endgroup$ Jul 15, 2015 at 17:00

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If I understand your question correctly, the answer is no. For instance if $\sigma=(12)(34)$ in $S_4$, the cycle $(12)$ is not in the cyclic subgroup generated by $\sigma$.

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  • $\begingroup$ I think I began to realize this last night- but thank you for your confirmation. Could you possibly explain the correlation between orbits and disjoint cycles? The wikipedia article on the subject alludes to this, but I could not fully understand it. $\endgroup$ Jul 15, 2015 at 18:18
  • $\begingroup$ I think typically if you have some group of permutations on a set $X$ (possibly a subgroup of $S_n$), the orbit of an element of $X$ is the set of elements it maps to under the group of permutations. For the subgroup generated by a single permutation, the orbit of an element is simply the set of elements in its cycle. For instance for the subgroup $\langle (123)(4567)\rangle$, the orbit of $5$ is $\{4,5,6,7\}$. $\endgroup$
    – paw88789
    Jul 15, 2015 at 18:30

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