Groups in which every Lagrange subset is a factor

TL;DR

The paper investigates finite groups for which every Lagrange subset is a factor, a property called the strong CFS property. It develops notions of subset products $AB$, left/right/two-sided factors, and the CFS framework, and proves that the strong CFS property is preserved under subgroups via a key lemma. It then provides a complete classification: the strong CFS property holds exactly for the trivial group, cyclic groups of prime order, and the groups $C_2^2$, $C_4$, $C_2^3$, and $C_3^2$. This yields a sharp structural characterization and links subset tiling ideas to factorization properties in finite groups, clarifying which groups admit universal factorability of Lagrange subsets.

Abstract

We determine the finite groups $G$ in which every subset $A \subseteq G$ of cardinality dividing the order of $G$ is a \emph{factor}, i.e. has a complement $B \subseteq G$ of cardinality $|G|/|A|$ such that $G = A \cdot B$ or $G = B \cdot A$.

Theorems & Definitions (8)

• Definition 1.1
• Lemma 1.2
• proof
• Lemma 1.3
• proof
• Theorem 2.1
• proof
• Corollary 2.2