Kourovka Notebook Problem 21.120

Definitions and mathematical statement

Fix a prime $p$. A pro-$p$ group is an inverse limit of finite $p$-groups, with its inverse-limit topology. The pro-$p$ completion of an abstract group $H$ is $\widehat H_p=\varprojlim H/N$, where $N$ runs over normal subgroups of finite $p$-power index.

A pro-$p$ group is strictly finitely presented if it is isomorphic to $\widehat H_p$ for some finitely presented abstract group $H$. In the relative version, $H$ need only be finitely presented within a finitely based variety of groups. Such a variety is specified by finitely many group identities. A finite presentation within it means a quotient of a relatively free group on finitely many generators by the normal closure of finitely many relators.

A pro-$p$ group $G$ is finitely axiomatizable among pro-$p$ groups if there is one parameter-free first-order sentence $\sigma$, using multiplication, inverse, identity, and equality, such that $G\models\sigma$ and every pro-$p$ group satisfying $\sigma$ is isomorphic to $G$ as a topological group. Quantifiers in the sentence range over elements, not over open subsets.

(a) Does there exist a strictly finitely presented pro-$p$ group that is not finitely axiomatizable in this class? Ask the same question with "strictly" replaced by "relatively strictly".

(b) Is every finitely generated free pro-$p$ group finitely axiomatizable? The free pro-$p$ group on $d$ generators is the pro-$p$ completion of the abstract free group of rank $d$; maps from its generators into any pro-$p$ group extend uniquely to continuous homomorphisms.

Short English statement

Can strict or relative finite presentability fail to give a single first-order characterization among pro-$p$ groups, and are free pro-$p$ groups finitely axiomatizable?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2629 (KOU-21.120), Kourovka Notebook - New Problems, Issue 21. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 The Kourovka Notebook, 21st edition, new problems, Problem 21.120. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.
• S3 D. Segal, On some relatively free pro-p groups; framework for relative presentations and finite axiomatizability. https://arxiv.org/abs/2505.04816.