On families of nilpotent subgroups and associated coset posets

TL;DR

This paper investigates the topological properties of coset posets related to subgroups of nilpotent groups, establishing conditions for simple connectivity and homotopy types for specific groups.

Contribution

It characterizes the homotopy types of coset posets for certain nilpotent groups and connects these properties to group-theoretic conditions like being 2-Engel.

Findings

Coset poset is simply-connected iff the group is 2-Engel.

Coset poset is 2-connected iff the group is nilpotent of class ≤ 2.

Determined homotopy type for specific matrix and Burnside groups.

Abstract

We study some properties of the coset poset associated with the family of subgroups of class $\leq 2$ of a nilpotent group of class $\leq 3$. 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 $3$.