Higher commutativity and nilpotency in finite groups

TL;DR

This paper explores the structure and properties of nilpotent subgroups within finite groups, introducing new poset-based invariants and connecting them to topological and probabilistic aspects of group theory.

Contribution

It introduces a family of finite Dirichlet series associated with nilpotent subgroups of finite groups and links these to topological invariants and classifying space filtrations.

Findings

Defined posets of nilpotent subgroups of bounded class

Established connections between Dirichlet series and topological invariants

Analyzed probabilistic properties related to subgroup structures

Abstract

We consider ordered tuples in finite groups generating nilpotent subgroups. Given an integer $q$ we consider the poset of nilpotent subgroups of class less than $q$ and its corresponding coset poset. These posets give rise to a family of finite Dirichlet series parametrized by the nilpotency class of the subgroups, which in turn reflect probabilistic and topological invariants determined by these subgroups. Connections of these series to filtrations of classifying spaces of a group are discussed.