Profinite groups with many elements with large nilpotentizer and generalizations

TL;DR

This paper explores the structure of profinite groups by analyzing elements with large nilpotentizers relative to families of finite groups, revealing how these elements influence the overall group structure.

Contribution

It introduces a framework for studying elements with large nilpotentizers in profinite groups based on families of finite groups, extending understanding of their structural properties.

Findings

Identification of conditions under which elements have large nilpotentizers

Relationship between nilpotentizer size and group structure

Generalizations for various families of finite groups

Abstract

Given a profinite group $G$ and a family $\mathcal{F}$ of finite groups closed under taking subgroups, direct products and quotients, denote by $\mathcal{F}(G)$ the set of elements $g \in G$ such that $\{x \in G\ |\ \langle g,x \rangle \ \mbox{is a pro-}\mathcal{F} \mbox{ group}\}$ has positive Haar measure. We investigate the properties of $\mathcal{F}(G)$ for various choices of $\mathcal{F}$ and its influence on the structure of $G$.