Profinite groups with many elements with large nilpotentizer and generalizations

TL;DR

This work analyzes how a Haar-measurable notion of generating a pro-\\F subgroup, via $\\mathcal{F}_G(x)$ and $\\mathcal{F}(G)$, reflects the global structure of a profinite group. It establishes a dichotomy: for a broad class of finite-group families $\\mathcal{F}$, $\\mathcal{F}(G)$ is not guaranteed to be closed unless $\\mathcal{F}$ is the class of all finite groups, with explicit counterexamples and a bounded-exponent caveat. It then specializes to solvable and nilpotent contexts, proving that positive measure of solvabilizers or nilpotentizers forces strong global structure such as virtual prosolvability or virtual pronilpotence, and derives local and p-local refinements. The paper also shows sharp limitations by constructing counterexamples to stronger conjectures involving pronilpotency-like sets $\\Lambda_G(x)$ and their p-local variants, clarifying when measure-positivity implies global algebraic constraints. Overall, it clarifies the extent to which measure-theoretic properties dictate the profinite group's architecture and where local data fail to control global structure.

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 $G$.

Theorems & Definitions (43)

• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Theorem 9
• Theorem 10
• Theorem 11
• Theorem 12
• proof : Proof of \ref{['allfingrps']}