Pro-p groups with constant generating number on open subgroups

TL;DR

This paper classifies finitely generated pro-p groups with a constant minimal number of generators across all open subgroups, addressing a question in p-adic analytic group theory.

Contribution

It provides a complete classification of such pro-p groups, especially within p-adic analytic groups, answering a longstanding question of Iwasawa.

Findings

Identified conditions for pro-p groups with constant generator number

Characterized these groups within p-adic analytic categories

Extended understanding of subgroup generation properties

Abstract

Let p be a prime. We classify finitely generated pro-p groups G which satisfy d(H) = d(G) for all open subgroups H of G. Here d(H) denotes the minimal number of topological generators for the subgroup H. Within the category of p-adic analytic pro-p groups, this answers a question of Iwasawa.