论文标题
$ 1 $ - 产品在非亚伯群体中的一致条件
$1$-product problems with congruence conditions in nonabelian groups
论文作者
论文摘要
令$ g $为有限的组，$ d_ {2n} $是$ 2N $元素的二面体组。对于积极的整数$ d $，令$ \ mathsf {s} _ {d \ mathbb {n}}}（g）$表示最小的整数$ \ ell \ in \ in \ mathbb {n} _0 _0 \ cup \ cup \ cup \ cup \ {+f {+f {+\ iffty \} $ $ | t | \ equiv 0 $（mod $ d $）的非发行$ 1 $ - 产品$ t $ t $。在本文中，我们主要研究二面体组$ d_ {2n} $的问题，并确定其确切值：$ \ m athsf {s} _ {d \ mathbb {n}}}（d_ {2n}）= 2d+\ lfloor log_2n \ rfloor $，如果$ \ mathsf {s} _ {d \ mathbb {n}}（d_ {2n}）= nd+1 $，如果$ gcd（n，d）= 1 $。此外，我们还分析了MetacyClic组的问题$ C_P \ ltimes_s c_q $并获得结果：$ \ Mathsf {s} _ {kp \ Mathbb {n}}（c_p \ ltimes_s c_q）= lcm（kp，q） $ P | Q-1 $。
Let $G$ be a finite group and $D_{2n}$ be the dihedral group of $2n$ elements. For a positive integer $d$, let $\mathsf{s}_{d\mathbb{N}}(G)$ denote the smallest integer $\ell\in \mathbb{N}_0\cup \{+\infty\}$ such that every sequence $S$ over $G$ of length $|S|\geq \ell$ has a nonempty $1$-product subsequence $T$ with $|T|\equiv 0$ (mod $d$). In this paper, we mainly study the problem for dihedral groups $D_{2n}$ and determine their exact values: $\mathsf{s}_{d\mathbb{N}}(D_{2n})=2d+\lfloor log_2n\rfloor$, if $d$ is odd with $n|d$; $\mathsf{s}_{d\mathbb{N}}(D_{2n})=nd+1$, if $gcd(n,d)=1$. Furthermore, we also analysis the problem for metacyclic groups $C_p\ltimes_s C_q$ and obtain a result: $\mathsf{s}_{kp\mathbb{N}}(C_p\ltimes_s C_q)=lcm(kp,q)+p-2+gcd(kp,q)$, where $p\geq 3$ and $p|q-1$.
