On some relatively free pro-p groups

TL;DR

This paper proves that certain relatively free pro-p groups with specific structural properties are finitely axiomatizable within the class of all profinite groups, highlighting their logical definability.

Contribution

It establishes finite axiomatizability for specific relatively free pro-p groups, advancing understanding of their logical and algebraic structure.

Findings

Relatively free centre-by-metabelian pro-p groups are finitely axiomatizable.

Class-2 nilpotent-by-abelian pro-p groups are finitely axiomatizable.

Results apply to groups on 2 generators.

Abstract

It is shown that the relatively free centre-by-metabelian and (class-2 nilpotent)-by-abelian pro-p groups on 2 generators are each finitely axiomatizable in the class of all profinite groups.