An axiomatizable profinite group with infinitely many open subgroups of index 2

TL;DR

This paper proves that a profinite group sharing the same first-order theory as a specific infinite product of dihedral groups is necessarily isomorphic to that product, highlighting a strong model-theoretic characterization.

Contribution

It establishes an axiomatization and uniqueness result for a class of profinite groups based on their first-order theory.

Findings

Profinite group with same first-order theory as the product of dihedral groups is isomorphic to it.

Provides an axiomatization for this class of profinite groups.

Shows the uniqueness of the group structure via model-theoretic properties.

Abstract

We show that a profinite group with the same first-order theory as the direct product over all odd primes $p$ of the dihedral group of order $2p$, is necessarily isomorphic to this direct product.