replace the item in the th position by the one that was in the th place. This action may be written as the rule . Now if we act on this by another permutation then we need to first relabel the items by writing . Then takes this to This proves that the action is a left action: . Now we consider the second interpretation of the action of , which is the opposite of the first. The following descriptions of the second interpretation are all equivalent: For each , move the item in the th place to the th place. For each , move the item in the th place to the th place. For each , replace the item in the th position by the one that was in the th place. This action may be written as the rule . In order to act on this by another permutation , again we first relabel the items by writing . Then the action of takes this to This proves that our second interpretation of the action is a right action: . Example If is the 3-cycle and is the transposition , then since we write maps on the left of their arguments we have Using the first interpretation we have , the result of which agrees with the action of on . So . On the other hand, if we use the second interpretation, we have , the result of which agrees with the action of on . So . Maps written on the right Sometimes people like to write maps on the right of their arguments. This is a convenient convention to adopt when working with symmetric groups as diagram algebras, for instance, since then one may read compositions from left to right instead of from right to left. The question is: how does this affect the two interpretations of the place-permutation action of a symmetric group? The answer is simple. By writing maps on the right instead of on the left we are reversing the order of composition, so in effect we replace by its opposite group . This is the same group, but the order of compositions is reversed. Reversing the order of compositions evidently changes left actions into right ones, and vice versa, changes right actions into left ones. This means that our first interpretation becomes a right action while the second becomes a