Some constructions of factorizations of symmetric group

Let B1, . . ., Bk be the subsets of a finite non-abelian group G such that G = B1 . . . Bk, where k ≥ 2. If |G| = |B1|. . .Bk|, or equivalently, the group multiplication map B1 ×. . .× Bk → G is a bijection, then G = B1 . . . Bk is a k-fold factorization of G. Let Sn and An be the symmetric group and the alternating group of degree n respectively. We show some constructions of k-fold factorizations of Sn involving Sn−1 and An, where k = 2, 3, . . ., n− 1. In addition, the m-th power of the permutation (1, 2, . . ., n) is studied to form the elements in the factorization subsets, for 2 ≤ m ≤ n− 1.