Factorizations of finite groups

2021/02/17 by Kabenyuk, Mikhail
#20D60 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2102.08605

Abstract

A finite group G is called k-factorizable if for any factorization |G|=a₁⋯ a_k with a_i>1 there exist subsets A_i of G with |A_i|=a_i such that G=A₁⋯ A_k. The main results are as follows. 1. For every integer k≥3 there exist a finite group G such that G is not k-factorizable. 2. If a Sylow 2-subgroup of a finite group G is elementary abelian, all involutions of G are conjugate, and the centralizer of every involution is the direct product of a Sylow 2-subgroup of G and a group of odd order. Then G is not 3-factorizable. 3. We give a complete list of not k-factorizable groups for some k of order at most 100.

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