On families of nilpotent subgroups and associated coset posets

IF 0.5 4区数学

Journal of Homotopy and Related Structures Pub Date : 2022-09-14 DOI:10.1007/s40062-022-00315-w

Simon Gritschacher, Bernardo Villarreal

引用次数: 0

Abstract

We study some properties of the coset poset associated with the family of subgroups of class \(\le 2\) of a nilpotent group of class \(\le 3\). 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 \(4\times 4\) upper unitriangular matrices over \(\mathbb {F}_p\), and for the Burnside groups of exponent 3.

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幂零子群的族及相关的陪集集

研究了一类幂零群\(\le 3\)的类\(\le 2\)的子群族的协集偏序集的一些性质。证明了在群的某些假设下，当且仅当群是2- engel，当且仅当群是2类或更小的幂零时，群的余集偏序集是单连通的;我们确定了\(\mathbb {F}_p\)上的\(4\times 4\)上单角矩阵群和指数为3的Burnside群的余集偏序的同伦类型。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Homotopy and Related Structures Mathematics-Geometry and Topology

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期刊介绍： Journal of Homotopy and Related Structures (JHRS) is a fully refereed international journal dealing with homotopy and related structures of mathematical and physical sciences. Journal of Homotopy and Related Structures is intended to publish papers on Homotopy in the broad sense and its related areas like Homological and homotopical algebra, K-theory, topology of manifolds, geometric and categorical structures, homology theories, topological groups and algebras, stable homotopy theory, group actions, algebraic varieties, category theory, cobordism theory, controlled topology, noncommutative geometry, motivic cohomology, differential topology, algebraic geometry.