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

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).