The theory of $(\lambda ,\mu )$-fold factorizations of groups

Motivated by the recent study of $\lambda$-fold near-factorizations of finite groups by Kreher, Li, and Stinson (Designs, Codes and Cryptography, 2026), we introduce and systematically investigate $(\lambda ,\mu )$-fold factorizations of finite groups, with particular emphasis on the abelian case. These structures form a natural relaxation of $\lambda$-fold near-factorizations and encompass several related families that have previously been studied in different contexts. We establish the foundational theory of $(\lambda ,\mu )$-fold factorizations, provide two general constructions, and prove several nonexistence results for finite abelian groups.