Near-factorizations of dihedral groups
Donald L. Kreher, Maura B. Paterson, Douglas R. Stinson
TL;DR
The paper investigates near-factorizations of nonabelian groups with a focus on dihedral groups, clarifying when known constructions yield equivalent near-factorizations and providing new nonequivalent examples. It connects near-factorizations to SEDFs/GSEDFs, analyzes two classic dihedral NF constructions (CGHK and BHS) for equivalence, and explores the Pêcher transform that transfers symmetric NF from cyclic groups to strongly symmetric NF in dihedral groups, including an inverse transform. Through explicit examples in $D_{41}$, $D_{95}$, and products of dihedral or cyclic groups, the authors demonstrate the existence of nonequivalent NF in several nonabelian settings and study when the Pêcher-type method preserves equivalence, aided by numerical gcd conditions. The work advances understanding of NF structure in nonabelian groups, illustrates the limitations of certain equivalence criteria, and raises open questions about gcd constraints and broader nonabelian/noncyclic abelian cases.
Abstract
We investigate near-factorizations of nonabelian groups, concentrating on dihedral groups. We show that some known constructions of near-factorizations in dihedral groups yield equivalent near-factorizations. In fact, there are very few known examples of nonequivalent near-factorizations in dihedral or other nonabelian groups; we provide some new examples with the aid of the computer. We also analyse a construction for near-factorizations in dihedral groups from near-factorizations in cyclic groups, due to Pêcher, and we investigate when nonequivalent near-factorizations can be obtained by this method.