arXivDaily arXiv每日学术速递 周一至周五更新
arXiv周末暂无论文更新，休息一下吧，周末愉快~~
arXiv 1402.0799math.GR
收藏

断面作为有限生成群中的生成集

Transversals as generating sets in finitely generated groups

Jack Button, Maurice Chiodo, Mariano Zeron-Medina Laris

备注：
15 pages, 6 figures. This is the version submitted for publication
发表信息：
Bulletin of the Australian Mathematical Society 93 (2015) 47-60
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

研究了有限生成群的有限指数子群的断面。证明了当$H$是秩$n$群$G$的子群且$H$在$G$中的索引至少为$n$时，我们可以构造$H$的左截线，它包含$G$的生成集的大小为$n$，并且当$G$被证明时，构造是算法的.我们还表明，在$G$有秩$n \leq3$的情况下，有一个同时左右横向的$H$，其中包含一个生成集的大小$n$为$G$。最后，我们表明，如果$H$是一个小组的秩$n$组$G$的索引小于$3 \cdot 2^{n-1}$，$H$不包含本原元的$G$，那么$H$是正常的$G$和$G/H \cong C_{2}^{n}$。

英文摘要：

We explore transversals of finite index subgroups of finitely generated groups. We show that when $H$ is a subgroup of a rank $n$ group $G$ and $H$ has index at least $n$ in $G$ then we can construct a left transversal for $H$ which contains a generating set of size $n$ for $G$, and that the construction is algorithmic when $G$ is finitely presented. We also show that, in the case where $G$ has rank $n \leq3$, there is a simultaneous left-right transversal for $H$ which contains a generating set of size $n$ for $G$. We finish by showing that if $H$ is a subgroup of a rank $n$ group $G$ with index less than $3 \cdot 2^{n-1}$, and $H$ contains no primitive elements of $G$, then $H$ is normal in $G$ and $G/H \cong C_{2}^{n}$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑