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

Theorem 1.1.

The profinite group G given by the direct product of the dihedral groups Dp over all odd primes p, is axiomatizable.

Definition 2.1.

For a profinite group Γ we set

and note that this defines a group if Aut⁢(Γ) is abelian.

Definition 2.2.

For a prime number p we denote by Cp the cyclic group of order p. If p is odd, then Aut⁢(Cp)≅Cp-1 so Inv⁢(Cp) is a group isomorphic to C2. The semidirect product Cp⋊Inv⁢(Cp) is the dihedral group Dp. Let ρp be a generator of Cp, and let ϵp be a generator of Inv⁢(Cp) so that they generate Dp and we have ϵp⁢ρp⁢ϵp=ρp-1.

Lemma 2.3.

For an odd prime p we have Cp={[a,b]∣a,b∈Dp}.

Proof.

For one inclusion note that Dp/Cp is abelian, and for the other one take some ρpn∈Cp. As p is odd, there exists a k∈ℤ such that 2⁢k≡n⁢(p). We find that [ϵp,ϵp⁢ρpk]=ϵp⁢ϵp⁢ρpk⁢ϵp⁢ρp-k⁢ϵp=ρpk⁢ϵp⁢ρp-k⁢ϵp=ρpk⁢ρpk=ρp2⁢k=ρpn. ∎

Remark 2.4.

A similar argument shows that Cp≤Dp is its own centralizer.

Definition 2.5.

We set G:=∏Dp, C:=∏Cp and E:=∏〈ϵp〉, where the products (here and in the sequel) are always taken over all odd primes p.

Definition 2.6.

For a profinite group Γ we denote by Γ′ its profinite commutator, which is the closed subgroup of Γ generated by {[a,b]∣a,b∈Γ}.

Remark 2.7.

Let us present some more first-order sentences that hold in G.

1. By (2.1), (2.3) the following first-order sentence is valid in G

   (2.5)∀g⁢∃h,k:g2=[h,k].

2. It follows from Remark 2.4 that C is its own centralizer in G. In view of (2.3), this is tantamount to the following first-order sentence

   (2.6)∀x((∀y,z:x[y,z]x-1=[y,z])→∃a,b:x=[a,b]).

3. Fix an odd prime p. We can think of ϵp,ρp as elements of E,C respectively, and thus also as elements of G. The first-order sentence

   (2.7)∃x⁢∀y,z:x⁢[y,z]⁢x-1=[y,z]-1↔[y,z]p=1

   holds in G since we can take x=ϵp.

Definition 2.8.

We say that profinite groups Γ,Λ are elementarily equivalent if they have the same first-order theory, and denote this by Γ≡Λ.

Remark 3.1.

From our proof of Theorem 1.1 one can extract an explicit (infinite) set of axioms characterizing G up to an isomorphism. For that, one just has to find a set of first-order sentences characterizing C, and such a set can be obtained from the proof of [4, Theorem A]. Let us however write down such axioms for C explicitly.

First, we take a sentence saying that C is abelian. Since an abelian profinite group is isomorphic to the direct product of its Sylow subgroups, it suffices to include axioms assuring that the Sylow subgroups are elementary abelian, and cyclic. For that matter, include a sentence saying that C=C2, and fix an odd prime p. Finally, take a sentence saying that C/Cp2≅ℤ/p⁢ℤ, so that the p-Sylow subgroup is isomorphic to ℤ/p⁢ℤ.