is a group homomorphism ({+1, −1} is a group under multiplication, where +1 is e, the neutral element). The kernel of this homomorphism, that is, the set of all even permutations, is called the alternating group A n. It is a normal subgroup of S n, and for n ≥ 2 it has n!/2 elements. The group S n is the semidirect product of A n and any subgroup generated by a single transposition.