Alternating group A_n multifold-factorizability (Conjecture 6.1)

Establish that for every integer n > 5, the alternating group A_n is multifold-factorizable: for any factorization |A_n| = a_1 ··· a_k with k ≥ 2 and each a_i > 1, there exist subsets A_1, …, A_k ⊂ A_n with |A_i| = a_i such that every element g ∈ A_n has a unique representation g = a_1 ··· a_k with a_i ∈ A_i.

Background

The paper studies multifold factorizations of finite groups, proving that the simple groups of orders 168 (PSL(2,7)) and 360 (A_6) are multifold-factorizable. Multifold-factorizability requires that for any integer k ≥ 2 and any factorization of the group order into k integers greater than 1, there is a corresponding k-factorization of the group into subsets giving unique representations of elements.

Known counterexamples include A_4 and A_5 not being multifold-factorizable, while small symmetric groups S_n for 2 ≤ n ≤ 6 are multifold-factorizable. Motivated by these results, the authors propose extending multifold-factorizability to all alternating groups A_n with n > 5.