A classification of finite groups with small Davenport constant

TL;DR

This paper classifies finite groups according to their Davenport constant, a key invariant in additive combinatorics, providing new insights into the structure of groups with small or specific Davenport constants.

Contribution

It offers a classification framework for finite groups based on their Davenport constant, advancing understanding of the group's structure related to this invariant.

Findings

Identifies finite groups with specific Davenport constants.

Provides structural characterizations for groups with small Davenport constants.

Establishes finiteness results for groups with given Davenport constant.

Abstract

Let $G$ be a finite group. By a sequence over $G$, we mean a finite unordered string of terms from $G$ with repetition 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 $G$. Then, the 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. The Davenport constant is a combinatorial group invariant that has been studied fruitfully over several decades in additive combinatorics, invariant theory, and factorization theory, etc. Apart from a few cases of finite groups, the precise value of the Davenport constant is unknown. Even in the abelian case, little is known beyond groups of rank at most two. On the other hand, for a fixed positive integer $r$, structural results characterizing which groups $G$ satisfy $\mathsf D (G) = r$ are rare. We only know that there are finitely many such groups. In this paper, we study the classification of finite groups based on the Davenport constant.