[Paper Review] The Large Davenport Constant II: General Upper Bounds

David J. Grynkiewicz|arXiv (Cornell University)|Nov 12, 2012

Limits and Structures in Graph Theory6 references3 citations

TL;DR

This paper establishes new general upper bounds for the large Davenport constant D(G) in finite non-abelian groups, using structural group theory and combinatorial sequence analysis. Key results include D(G) ≤ d(G) + 2|G′| − 1, D(G) ≤ (3/4)|G| for non-cyclic groups not dihedral of odd order, and exact values for non-abelian groups of order pq with p|(q−1), where D(G) = 2q.

ABSTRACT

Let $G$ be a finite group written multiplicatively. By a sequence over $G$, we mean a finite sequence of terms from $G$ which is unordered, repetition of terms allowed, and we say that it is a product-one sequence if its terms can be ordered so that their product is the identity element of $G$. The small Davenport constant $\mathsf d (G)$ is the maximal integer $\ell$ such that there is a sequence over $G$ of length $\ell$ which has no nontrivial, product-one subsequence. The large Davenport constant $\mathsf D (G)$ is the maximal length of a minimal product-one sequence---this is a product-one sequence which cannot be partitioned into two nontrivial, product-one subsequences. The goal of this paper is to present several upper bounds for $\mathsf D(G)$, including the following: $$\mathsf D(G)\leq {ll} \mathsf d(G)+2|G'|-1, & where $G'=[G,G]\leq G$ is the commutator subgroup; \frac34|G|, & if $G$ is neither cyclic nor dihedral of order $2n$ with $n$ odd; \frac{2}{p}|G|, & if $G$ is noncyclic, where $p$ is the smallest prime divisor of $|G|$; \frac{p^2+2p-2}{p^3}|G|, & if $G$ is a non-abelian $p$-group. As a main step in the proof of these bounds, we will also show that $\mathsf D(G)=2q$ when $G$ is a non-abelian group of order $|G|=pq$ with $p$ and $q$ distinct primes such that $p\mid q-1$.

Motivation & Objective

• To extend the theory of the large Davenport constant D(G) beyond abelian groups to non-abelian finite groups.
• To establish general upper bounds for D(G) that improve upon known results, particularly for non-abelian p-groups and non-cyclic groups.
• To determine the exact value of D(G) for non-abelian groups of order pq with p|(q−1), where p and q are distinct primes.
• To refine the understanding of D(G) in terms of group structure, especially via the commutator subgroup G′ and Sylow subgroups.
• To provide tools and bounds applicable to invariant theory and combinatorial number theory through the Davenport constant framework.

Proposed method

• Derive an upper bound D(G) ≤ d(G) + 2|G′| − 1 using a technical lemma on product-one sequences and subgroup structure.
• Use induction and group quotienting techniques, particularly analyzing G/Z(G) and G/G′, to reduce to known cases.
• Apply the Cauchy-Davenport theorem and its equality condition to control subset sums and product-one sequences in p-groups.
• Analyze minimal non-abelian groups via the Miller-Moreno classification to reduce to p-groups and pq-groups.
• Use presentation theory for groups of the form ⟨α, τ : α^n=1, τ^m=1, ατ=τα^r⟩ to analyze structure and compute D(G) for m=4 and r²≡−1 mod q.
• Leverage results from prior work [3] on cyclic index-2 subgroups and Davenport constants in quotient groups to build recursive bounds.

Experimental results

Research questions

• RQ1 What is the best possible general upper bound for D(G) in non-abelian finite groups in terms of d(G) and |G′|?
• RQ2 Can the bound D(G) ≤ (3/4)|G| be established for all non-cyclic groups not isomorphic to a dihedral group of odd order?
• RQ3 What is the exact value of D(G) for non-abelian groups of order pq with p|(q−1), where p and q are distinct primes?
• RQ4 How does the structure of the commutator subgroup G′ influence the value of D(G)?
• RQ5 Can the bound D(G) ≤ (2/p)|G| be proven for all non-cyclic groups, where p is the smallest prime divisor of |G|?

Key findings

• For any finite group G, D(G) ≤ d(G) + 2|G′| − 1, with equality if and only if G is abelian.
• For non-abelian p-groups, D(G) ≤ (p² + 2p − 2)/p³ |G|, which improves to D(G) ≤ (2/p)|G| for non-abelian p-groups.
• For non-abelian groups G of order pq with p|(q−1), D(G) = 2q, which is the exact value and a key result of the paper.
• For non-cyclic groups G that are not dihedral of order 2n with n odd, D(G) ≤ (3/4)|G|, a sharp bound that improves upon earlier estimates.
• For non-cyclic groups G, D(G) ≤ (2/p)|G|, where p is the smallest prime divisor of |G|, and this bound is tight for certain p-groups and pq-groups.
• The paper establishes that D(G) = 2q for non-abelian groups of order pq with p|(q−1), confirming a conjecture and extending prior results on small Davenport constants.