imagine for a second that all symmetric groups were abelian, then is it true that every group would be abelian? (how cool would that be! )
It would mean all of group theory can be summarized in 60 pages.
Show for n>2
I just want to add, since NonCommAlg was implicit about this, this condition is necessary to have a third element. If n>=2 then writing (12)(23) is not allowed since there is no third element. That is why this condition is necessary.