On families of nilpotent subgroups and associated coset posets

被引：0

作者：

Simon Gritschacher

Bernardo Villarreal

机构：

[1] Ludwig-Maximilians-University,

[2] Centro de Investigación en Matemáticas,undefined

来源：

Journal of Homotopy and Related Structures | 2022年 / 17卷

关键词：

Nilpotent group; 2-Engel group; Colimit of groups; Coset poset; Higher generation; Simplicial set; Simplicial complex; Primary 57M07; 20F18; 55U10; Secondary 20F12; 20F45;

D O I：

暂无

中图分类号：

学科分类号：

摘要：

We study some properties of the coset poset associated with the family of subgroups of class ≤2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\le 2$$\end{document} of a nilpotent group of class ≤3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\le 3$$\end{document}. We prove that under certain assumptions on the group the coset poset is simply-connected if and only if the group is 2-Engel, and 2-connected if and only if the group is nilpotent of class 2 or less. We determine the homotopy type of the coset poset for the group of 4×4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$4\times 4$$\end{document} upper unitriangular matrices over Fp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_p$$\end{document}, and for the Burnside groups of exponent 3.

引用

页码：493 / 509

页数：16

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