Extremal Incomplete Sets in Finite Abelian Groups

Abstract

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