Extremal incomplete sets in finite abelian groups

TL;DR

This paper characterizes extremal subsets in finite abelian groups that fail to span the group, focusing on groups with even order or order product of two primes under certain conditions.

Contribution

It provides a complete characterization of extremal subsets of size critical number minus one in specific classes of finite abelian groups, extending previous results.

Findings

Characterization of extremal subsets for even order groups with |G| ≥ 36.

Characterization for groups where |G|=pq with primes p,q and q≥2p+3.

Extension of known results to new classes of finite abelian groups.

Abstract

Let $G$ be a finite abelian group. The critical number ${\rm cr}(G)$ of $G$ is the least positive integer $\ell$ such that every subset $A\subseteq G\setminus\{0\}$ of cardinality at least $\ell$ spans $G$, i.e., every element of $G$ can be written as a nonempty sum of distinct elements of $A$. The exact values of the critical number have been completely determined recently for all finite abelian groups. The structure of these sets of cardinality ${\rm cr}(G)-1$ which fail to span $G$ has also been characterized except for the case that $|G|$ is an even number and the case that $|G|=pq$ with $p,q$ are primes. In this paper, we characterize these extremal subsets for $|G|\geq 36$ is an even number, or $|G|=pq$ with $p,q$ are primes and $q\geq 2p+3$.