On families of nilpotent subgroups and associated coset posets

DOI10.1007/S40062-022-00315-WarXiv2104.06869MaRDI QIDQ6365302

Bernardo Villarreal, Simon Philipp Gritschacher

Publication date: 14 April 2021

Abstract: We study some properties of the coset poset associated with the family of subgroups of class $leq2$ of a nilpotent group of class $leq3$. 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 $4imes4$ upper unitriangular matrices over $mathbb{F}_p$, and for the Burnside groups of exponent $3$.

Mathematics Subject Classification ID

Nilpotent groups (20F18) Topological methods in group theory (57M07) Simplicial sets and complexes in algebraic topology (55U10) Commutator calculus (20F12) Engel conditions (20F45)

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