One of the problems in the last homework set is to study the derived group of the symmetric group .
Recall that if is a group and , then their commutator is defined as
The derived group is the subgroup of generated by the commutators.
Note that, since any permutation has the same parity as its inverse, any commutator in is even. This means that .
The following short program is Sage allows us to verify that, for , every element of is actually a commutator. The program generates a list of the commutators of , then verifies that this list is closed under products and inverses (so it is a group). It also lists the size of this group. Note that the size is precisely , so in these 4 cases: