On zero-sum problems over metacyclic groups Cn ⋊s C2

doi:10.1016/j.jcta.2026.106214

On zero-sum problems over metacyclic groups Cn ⋊s C2

2026-10-01

0

OA

AI

DOI：10.1016/j.jcta.2026.106214

Original

Save

View PDF

Abstract

En 中文

Let G be a finite group. A finite collection of elements from G, where the order is disregarded and repetitions are allowed, is said to be a product-one sequence if its elements can be ordered such that their product in G equals the identity element of G. Then, the Gao's constant E(G) of G is the smallest integer & ell; such that every sequence of length at least & ell; has a product-one subsequence of length |G|. For a positive integer n, we denote by C,, a cyclic group of order n. Let G = C,, & rtimes;a C2 with s2 = 1 (mod n) be a metacyclic group. The direct and inverse problems of E(G) were settled recently, except for the case that G = C3,,2 & rtimes;a C2 with n2 =/ 1, gcd(n2, 6) = 1, s = -1 (mod 3), and s = 1 (mod n2). In this paper, we complete the remaining case and hence for all metacyclic groups of the form G = C,,& rtimes;C2, the Gao's constant and the associated inverse problem are now fully settled (see Theorem 1.2). (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Keywords:

Product-one sequences

Gao's constant

Metacyclic groups

Zero-sum problems

On zero-sum problems over metacyclic groups Cn ⋊s C2

Abstract

AI Summary

Journal

Organization