A profinite analogue of Lasserre's theorem

TL;DR

This paper explores a profinite analogue of Lasserre's theorem, characterizing when a soluble pro-p group of finite rank is finitely axiomatizable within all profinite groups based on properties of its open subgroups.

Contribution

It establishes a criterion involving the finiteness of the image of the center in the abelianization of open subgroups for finite axiomatizability.

Findings

Characterization of finitely axiomatizable soluble pro-p groups

Connection between center images and finite axiomatizability

Conditions involving finite presentability and subgroup properties

Abstract

A soluble pro-p group of finite rank is finitely axiomatizable in the class of all profinite groups if and only if for each open subgroup H, the image of Z(H) in the abelianization of H is finite, subject to some suitable hypothesis of finite presentability.