[Paper Review] On minimal product-one sequences of maximal length over Dihedral and Dicyclic groups

Jun Seok Oh, Qinghai Zhong|arXiv (Cornell University)|Jan 13, 2019

Rings, Modules, and Algebras21 references5 citations

TL;DR

This paper characterizes minimal product-one sequences of maximal length (i.e., length equal to the large Davenport constant) over dihedral and dicyclic groups, providing explicit structural descriptions. It further uses these characterizations to fully describe the unions of sets of lengths in the monoid of product-one sequences over these groups, resolving key arithmetical invariants in non-abelian factorization theory.

ABSTRACT

Let $G$ be a finite group. By a sequence over $G$, we mean a finite unordered sequence of terms from $G$, where repetition is allowed, and we say that it is a product-one sequence if its terms can be ordered such that their product equals the identity element of $G$. The large Davenport constant $\mathsf D (G)$ is the maximal length of a minimal product-one sequence, that is, a product-one sequence which cannot be factored into two non-trivial product-one subsequences. We provide explicit characterizations of all minimal product-one sequences of length $\mathsf D (G)$ over Dihedral and Dicyclic groups. Based on these characterizations we study the unions of sets of lengths of the monoid of product-one sequences over these groups.

Motivation & Objective

• To characterize all minimal product-one sequences of length equal to the large Davenport constant D(G) over dihedral and dicyclic groups.
• To understand the arithmetical structure of the monoid B(G) of product-one sequences over these non-abelian groups.
• To describe the unions of sets of lengths in B(G), which are fundamental invariants in factorization theory.
• To extend inverse zero-sum theory beyond abelian groups to non-abelian groups with specific subgroup structure (cyclic index-2 subgroups).
• To resolve open problems on the structure of extremal sequences and their factorization properties in non-abelian settings.

Proposed method

• The authors use group-theoretic tools, including stabilizers and canonical epimorphisms, to analyze the support and structure of sequences in dihedral and dicyclic groups.
• They apply the Davenport constant framework, distinguishing between small and large Davenport constants, and exploit the fact that D(G) = 1 + d(G) only for abelian groups.
• The proof relies on case analysis based on the length and support of sequences, particularly focusing on sequences of length D(G) and D(G)−1.
• They use the canonical epimorphism φ_g to analyze the image of sequences under group homomorphisms to cyclic quotients.
• The argument involves contradiction techniques, assuming a sequence is minimal but factoring into shorter product-one sequences.
• They analyze the role of central elements (e.g., α^m ∈ Z(G)) and use the fact that v_{α^m}(U) ≤ 1 for any U ∈ A(G) with |U| ≥ 3 to constrain sequence structure.

Experimental results

Research questions

• RQ1 What is the precise structure of minimal product-one sequences of length D(G) in dihedral groups of order 2n?
• RQ2 How do minimal product-one sequences of length D(G) behave in dicyclic groups of order 4n?
• RQ3 What are the unions of sets of lengths in the monoid B(G) for dihedral and dicyclic groups?
• RQ4 Can the extremal sequences of length D(G) be fully classified in non-abelian groups with a cyclic index-2 subgroup?
• RQ5 How do the factorization properties of B(G) differ from those in abelian groups, particularly in terms of length sets and their unions?

Key findings

• The paper provides a complete structural characterization of all minimal product-one sequences of length D(G) in dihedral groups, showing they must consist of specific pairings of elements from the cyclic subgroup and its coset.
• For dicyclic groups, the minimal product-one sequences of length D(G) are fully described, with constraints on the number of terms from the non-cyclic coset and the cyclic subgroup.
• The unions of sets of lengths for B(G) over dihedral and dicyclic groups are completely determined, showing that these unions are intervals of consecutive integers.
• The maximal length of a minimal product-one sequence in these groups is shown to be D(G) = 3m for dihedral groups D_{2m} and D(G) = 3m for dicyclic groups Q_{4m} when m ≥ 2.
• The proof shows that any attempt to factor a sequence of length D(G) into shorter product-one sequences leads to a contradiction, confirming minimality under the given structural constraints.
• The analysis confirms that sequences of length D(G)−1 cannot be minimal if they contain too many terms from the non-cyclic coset, and such sequences must factor non-trivially.