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The Commutator Group of Symmetric Group

In a group, the product of two commutators need not be a commutator. Consequently the commutator group of a given group can not be defined as the set of all commutators, but only as the group generated by these. It seems there is no good criteria or investigations on the problem when all elements of the commutator group can be commutators. It shows that in the finite symmetric group S n all elements of its commutator group are commutators in theory. On this basis, the product of two commutators of S n is a commutator. In the following, the representation was given to the elements of commutator group [ S n·S n ]to be expressed by commutators. A counter example was used to show that such a group G existed and its elements contained in commutator group can not be represented by commutators.