On zero-sum problems over metacyclic groups $C_n{⋊}_s{C}_2$

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 $G=C_n{⋊}_s{C}_2$ with $s^2\equiv 1(\mathrm{mod}n)$ be a metacyclic group. The direct and inverse problems of $\mathsf{E}(G)$ were settled recently, except for the case that $G=C_{3n_2}{⋊}_s{C}_2$ with $n_2\ne 1$, $\gcd (n_2,6)=1$, $s\equiv -1(\mathrm{mod}3)$, and $s\equiv 1(\mathrm{mod}{n}_2)$. In this paper, we complete the remaining case and hence for all metacyclic groups of the form $G=C_n⋊C_2$, the Gao's constant and the associated inverse problem are now fully settled (see Theorem 1.2).

To appear in J. Combin. Theory Ser. A