An answer regarding factorizations of finite groups

2026/07/21 by Ryan McCulloch
Mathematics · #math.GR

paper · pdf

Abstract

In Question 19.35 of the Kourovka Notebook, M. H. Hooshmand asked whether, given a finite group G and a factorization |G| = n₁ ⋯ n_k, one can always find subsets A₁, … , A_k of G with |A_i| = n_i such that G = A₁ ⋯ A_k. G. M. Bergman provided a counterexample for k=3, and asked whether counterexamples exist for larger k. In this note we provide counterexamples for all k ≥ 3. The question is still open for k=2.

Related

• Factorizations of finite groups
• Complete factorizations of finite groups
• Factorizations of simple groups of order 168 and 360
• Factorizations of groups of small order
• Factorizations in finite groups