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

TL;DR

This paper determines Gao's constant for all metacyclic groups of the form $C_n times C_2$, completing previous results and solving the remaining case.

Contribution

It completes the calculation of Gao's constant for all such metacyclic groups, resolving the last open case in the problem.

Findings

Gao's constant $ ext{E}(G)$ is now known for all groups of the form $C_n times C_2$.

The paper solves the remaining open case where $G=C_{3n_2} times_s C_2$ with specific conditions.

The inverse problem associated with Gao's constant is also fully settled for these 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).