Factorizations of groups of small order

2021/03/15 by M. I. Kabenyuk, Kabenyuk, Mikhail
Engineering · Mathematics · Neuroscience · #20D60 #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Nuclear Receptors and Signaling #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2103.08353

openalex publication_date 2021/03/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be a finite group and let A₁,…,A_k be a collection of subsets of G such that G=A₁… A_k is the product of all the A_i's with |G|=|A₁|…|A_k|. We write G=A₁⋅…⋅ A_k and call this a k-fold factorization of G of the form (|A₁|,…,|A_k|) or more briefly an (|A₁|,…,|A_k|)-factorization of G. Let k≥2 be a fixed integer. If G has an (a₁,…,a_k)-factorization, whenever |G|=a₁… a_k with a_i>1, i=1,…,k, we say that G is k-factorizable. We say that G is multifold-factorizable if G is k-factorizable for any possible integer k≥2. In this paper we prove that there are exactly 6 non-multifold-factorizable groups among the groups of order at most 60. Here is their complete list: A₄, (C₂× C₂)\rtimes C₉, A₄× C₃, (C₂× C₂× C₂)\rtimes C₇, A₅, A₄× C₅. Some related open questions are presented.

Related

• Factorizations of simple groups of order 168 and 360
• Factorizations of finite groups
• An answer regarding factorizations of finite groups
• Complete factorizations of finite groups
• Factorization of finite simple groups