$λ$-fold near-factorizations of groups

Abstract

We initiate the study of $λ$-fold near-factorizations of groups with $λ> 1$. While $λ$-fold near-factorizations of groups with $λ= 1$ have been studied in numerous papers, this is the first detailed treatment for $λ> 1$. We establish fundamental properties of $λ$-fold near-factorizations and introduce the notion of equivalence. We prove various necessary conditions of $λ$-fold near-factorizations, including upper bounds on $λ$. We present three constructions of infinite families of $λ$-fold near-factorizations, highlighting the characterization of two subfamilies of $λ$-fold near-factorizations. We discuss a computational approach to $λ$-fold near-factorizations and tabulate computational results for abelian groups of small order.

Theorems & Definitions (80)

• Example 1.1
• Lemma 1.2
• Theorem 1.3
• proof
• Theorem 1.4
• proof
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• proof