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I know the following statement: Two cycles $\sigma, \tau \in S_n$ statisfies $\sigma \tau = \tau \sigma$ if and only if they have disjoint cycles or $\sigma$ is a power of $\tau$.

My question is: Is there exists a similar statement for two permutations (generalizing the previous) ?

It is clear that if two permutations are disjoint then they commute, but that is a sufficient condition. I ask about a "if and only if" condition. I know also the "iff" condition if we write the both permutations with disjoint cycles using the formula: $$ \sigma ( a_1 \ \cdots \ a_k ) \sigma ^{-1} = ( \sigma(a_1) \ \cdots \ \sigma(a_k) ) $$

My question is to find a more characterizing statement like the first one, but not only for cycles.

Lazare
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    It’ll be difficult; for example, the permutations $(12)(34)$, $(13)(24)$, and $(14)(23)$ pairwise commute, but given any two, neither of their component cycles are disjoint with either of the component cycles of the other, nor are the cycles powers of the same permutation. – Arturo Magidin Aug 14 '20 at 23:27
  • Yes indeed it's difficult. Thanks for your answer. Maybe certains conditions exists in link with other fields in maths, so if somone have an idea, don't hesitate. – Lazare Aug 15 '20 at 08:57
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    The permutations commute iff the orbits for the group generated by $\sigma$ and $\tau$ "look like" a discrete 2d torus as in this picture https://www.researchgate.net/figure/A-discrete-torus-and-a-fragment-of-the-picture-Torus-and-Sphere-by-Tom-Wilcox-2008-as_fig4_313807438 The meridians would be say the cycles of $\sigma$ and the parallels would be the cycles of $\tau$. – Abdelmalek Abdesselam Aug 20 '20 at 21:49
  • Thank you for your answer very interesting. Is there a geometric theory for commuting behind this point of view? I mean specific algebraic topology theorems? – Lazare Aug 22 '20 at 08:33
  • An answer given here https://math.stackexchange.com/questions/440568/are-there-any-conditions-such-that-2-permutations-in-s-4-are-commutative

    and an example https://math.stackexchange.com/questions/245990/find-all-permutations-that-commute-with-omega-1-9-7-10-12-2-54-113-6-8

     – Harshavardhan Aug 06 '21 at 06:56

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