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.