Homomorphic Stability of Finite Groups

Abstract

Let G be a group. The subsets A1, . . . , A k of G form a complete factorization of group G if if they are pairwise disjoint and each element g ∈ G is uniquely represented as g = a1 . . . a k , with ai ∈ Ai. We prove the following theorem.Theorem. Let G be a finite nilpotent group. If |G| = m1 . . . m k where m1, . . . , m k are integers greater 1 and k > 2, then there exist subsets A1, . . . , A k of G which form a complete factorization of group G and |Ai| = mi for all i = 1, 2, . . . , k.