Factorizations of finite groups

TL;DR

This paper investigates the concept of $k$-factorizability in finite groups, establishing existence results, specific structural conditions preventing certain factorizations, and classifying small groups that are not $k$-factorizable.

Contribution

It introduces the notion of $k$-factorizability, proves the existence of non-$k$-factorizable groups for all $k\,geq 3$, and classifies small groups failing to be $k$-factorizable.

Findings

For each $k\,geq 3$, there exists a finite group not $k$-factorizable.

Certain structural conditions prevent specific factorizations in groups of order $4m$.

Only 8 groups of order at most 100 are not $k$-factorizable for some $k$.

Abstract

A finite group $G$ is called $k$-factorizable if for every ordered factorization $|G|=a_1\cdots a_k$ into integers each greater than $1$ there exist subsets $A_1,\dots,A_k\subseteq G$ such that $|A_i|=a_i$ for each $i$ and $G=A_1\cdots A_k$. The main results are as follows. 1. For every integer $k\geq3$ there exists a finite group $G$ such that $G$ is not $k$-factorizable. 2. Let $G$ be a finite group of order $4m$. If a Sylow $2$-subgroup of $G$ is elementary abelian, all involutions of $G$ are conjugate, and the centralizer of every involution has a normal Sylow $2$-subgroup, then $G$ has no factorization of the form $G=ABC$ with $|A|=|C|=2$ and $|B|=m$. 3. Only $8$ groups of order at most $100$ fail to be $k$-factorizable for some $k$.