Prove that is generated by the set of all 3-cycles for .

Let and let be the set of 3-cycles in .

Note that contains , so that .

Recall that consists of all permutations which can be written as an even product of transpositions; more specifically, is generated by the set of all products of two distinct 2-cycles. (Here we use the fact that .) Each product of two 2-cycles has one of two forms.

If , note that .

If , note that .

Thus , hence .