
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 is called -factorizable if for every ordered factorization into integers each greater than there exist subsets such that for each and . The main results are as follows. 1. For every integer there exists a finite group such that is not -factorizable. 2. Let be a finite group of order . If a Sylow -subgroup of is elementary abelian, all involutions of are conjugate, and the centralizer of every involution has a normal Sylow -subgroup, then has no factorization of the form with and . 3. Only groups of order at most fail to be -factorizable for some .
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