Set-Direct Factorizations of Groups

TL;DR

This paper introduces a new concept of set-direct factorizations in groups, characterizes when such factorizations exist, and shows that simple groups do not admit non-trivial factorizations.

Contribution

It generalizes the notion of direct product decompositions to subsets of groups and provides a complete characterization for their existence.

Findings

A group has a set-direct factorization iff it is a central product with specific abelian conditions.

Simple groups do not admit non-trivial set-direct factorizations.

The paper analyzes special cases and provides illustrative examples.

Abstract

We consider factorizations $G=XY$ where $G$ is a general group, $X$ and $Y$ are normal subsets of $G$ and any $g\in G$ has a unique representation $g=xy$ with $x\in X$ and $y\in Y$. This definition coincides with the customary and extensively studied definition of a direct product decomposition by subsets of a finite abelian group. Our main result states that a group $G$ has such a factorization if and only if $G$ is a central product of $\left\langle X\right\rangle $ and $\left\langle Y\right\rangle $ and the central subgroup $\left\langle X\right\rangle \cap \left\langle Y\right\rangle $ satisfies certain abelian factorization conditions. We analyze some special cases and give examples. In particular, simple groups have no non-trivial set-direct factorization.