Originally Posted by

**Plato** This not intended to answer the original question. It is an interesting observation I think. Several years ago I saw this question posed a different way. *Prove that two permutations commute iff their active sets are disjoint*.

Example

$\displaystyle f = \left( {\begin{array}{*{20}c} 1 & 2 & 3 & 4 & 5 \\

1 & 4 & 3 & 5 & 2 \\\end{array}} \right)\quad \& \quad g = \left( {\begin{array}{*{20}c}

1 & 2 & 3 & 4 & 5 \\ 3 & 2 & 1 & 4 & 5 \\ \end{array}} \right)$.

The idea being that if an element is active in *f* it is not in *g*, and visa versa.