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Composition of Permutations

Let s and t be disjoint permutations in S_n. Show that st=ts.

Am I right in thinking that disjoint permutations are ones that don't have any of the same transpositions?

Also, can I just state that the transpositions can be done in any order, so you can do s first, then t or the other way round?
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nuodai
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A permutation is basically a shuffling up of a set (no new elements are introduced, no elements are removed, it's just a rearrangement of current elements); and two permutations are disjoint if one shuffles one set of things, and the other shuffles a completely different set of things (so s shuffles nothing that t shuffles, and t shuffles nothing that s shuffles). This isn't a formal definition by any means, but it should help you get to the answer. I recommend going about it by contradiction (i.e. assume that st≠tsst \ne ts and derive a contradiction).

Transpositions can't be done in any order. For example, (1 2 3)(4 5 6) is a product of two disjoint cycles and, in transposition notation, we have
(1 2 3)(4 5 6) = (1 3)(1 2)(4 6)(4 5)
...but that's not the same as (1 2)(1 3)(4 5)(4 6) for example.

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