Symmetric group S_n multifold-factorizability (Conjecture 6.2)

Establish that for every integer n ≥ 2, the symmetric group S_n is multifold-factorizable: for any factorization |S_n| = a_1 ··· a_k with k ≥ 2 and each a_i > 1, there exist subsets A_1, …, A_k ⊂ S_n with |A_i| = a_i such that every element g ∈ S_n has a unique representation g = a_1 ··· a_k with a_i ∈ A_i.

Background

The authors note that a positive resolution of Conjecture 6.1 (for A_n) would imply Conjecture 6.2 by Lemma 2.6(iii), which lifts factorizations from quotient groups to the group. They also remark that the converse is not true, citing that S_n is multifold-factorizable for 2 ≤ n ≤ 6, while A_4 and A_5 are not.

This conjecture complements the alternating-group conjecture and would, if proven, provide a broad class of non-abelian groups that are multifold-factorizable.