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.

中文翻译：

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关于幂零子群的族和相关陪集子集

我们研究了与类 \(\le 3\)的幂等群的类\ (\le 2\)的子群族相关的陪集偏序的一些性质。我们证明，在群的某些假设下，陪集偏序是单连通的当且仅当该群是 2-Engel 的，并且当且仅当该群是 2 类或更少的幂零时是 2-连通的。我们确定 \(\mathbb {F}_p\) 上\(\mathbb {F}_p\ ) 上的\(4\times 4\)上单位三角形矩阵群和指数 3 的 Burnside 群的陪集偏序的同伦类型。