On zero-sum problems over metacyclic groups $C_n \rtimes_s C_2$

TL;DR

This work resolves the Gao's constant ${\sf E}(G)$ and the inverse problem for all metacyclic groups $G=C_n\rtimes_s C_2$, completing the previously unsettled exceptional case $G=C_{3n_2}\rtimes_s C_2$ via the DeVos–Goddyn–Mohar additive-theory bound. It proves ${\sf E}(G)=9n_2$ in the exceptional case and ${\sf E}(G)=3n$ in the general case $G=C_n\rtimes_s C_2$, while also providing explicit extremal-sequence descriptions in the former and corresponding inverse results in both. The approach combines precise group-structural analysis with powerful additive combinatorics tools, extending non-abelian zero-sum theory to a broad class of metacyclic groups. This solidifies understanding of product-one subsequences in non-abelian settings and closes a long-standing gap in the direct and inverse problems for Gao's constant in metacyclic groups.

Abstract

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 $\mathsf E (G)$ 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 $C_n$ a cyclic group of order $n$. Let $G = C_n \rtimes_s C_2$ with $s^2\equiv 1\pmod n$ be a metacyclic group. The direct and inverse problems of $\mathsf E (G)$ were settled recently, except for the case that $G=C_{3n_2}\rtimes_s C_2$ with $n_2\neq 1$, $\gcd(n_2,6)=1$, $s\equiv -1 \pmod 3$, and $s\equiv 1\pmod {n_2}$. In this paper, we complete the remaining case and hence for all metacyclic groups of the form $G=C_n \rtimes C_2$, the Gao's constant and the associated inverse problem are now fully settled (see Theorem 1.2).

Theorems & Definitions (15)

• Theorem 1.1
• Theorem 1.2
• Remark 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• proof
• Lemma 3.1: DeVos-Goddyn-Mohar DGM or Gry
• Proposition 3.2
• proof