Title: On minimal product-one sequences of maximal length over the non-abelian group of order $pq$ Show Chinese title

Title: 关于非交换阶数为$pq$的群上最大长度的最小乘积一序列

Abstract: Let $G$ be a finite group. A sequence over $G$ is a finite multiset of elements of $G$, and it is called product-one if its terms can be ordered so that their product is the identity of $G$. The large Davenport constant $\D(G)$ is the maximal length of a minimal product-one sequence, that is, a product-one sequence that cannot be partitioned into two nontrivial product-one subsequences. Let $p,q$ be odd prime numbers with $p \mid q-1$ and let $C_q \rtimes C_p$ denote the non-abelian group of order $pq$. It is known that $\D(C_q \rtimes C_p) = 2q$. In this paper, we describe all minimal product-one sequences of length $2q$ over $C_q \rtimes C_p$. As an application, we further investigate the $k$-th elasticity (and, consequently, the union of sets containing $k$) of the monoid of product-one sequences over these groups. Show Chinese abstract

Abstract: 设 $G$ 是一个有限群。 在 $G$ 上的一个序列是 $G$ 元素的有限多重集合，如果其项可以排列使得它们的乘积是 $G$ 的单位元，则称为乘积为一的序列。 大戴维斯常数 $\D(G)$ 是一个极小乘积为一序列的最大长度，即不能被分割成两个非平凡乘积为一子序列的乘积为一序列。 设$p,q$为奇素数，且$p \mid q-1$，并让$C_q \rtimes C_p$表示阶为$pq$的非交换群。 已知 $\D(C_q \rtimes C_p) = 2q$。 在本文中，我们描述了在$C_q \rtimes C_p$上长度为$2q$的所有最小乘积为一的序列。 作为应用，我们进一步研究这些群上乘积为一序列的独异点的$k$-阶弹性（以及，从而，包含$k$的集合的并集）的弹性。