Elementary Equivalence of Profinite Groups

There are many examples of non-isomorphic pairs of finitely generated abstract groups that are elementarily equivalent. We show that the situation in the category of profinite groups is different: If two finitely generated profinite groups are elementarily equivalent (as abstract groups), then they are isomorphic. The proof applies a result of Nikolov and Segal which in turn relies on the classification of the finite simple groups. Our result does not hold any more if the profinite groups are not finitely generated. We give concrete examples of non-isomorphic profinite groups which are elementarily equivalent. MR Classification: 12E30 Directory: \Jarden\Diary\JL 13 March, 2008 * Supported by the Minkowski Center for Geometry at Tel Aviv University, established by the Minerva Foundation. ** Supported by the ISF and the Landau Center for Analysis at the Hebrew University of Jerusalem. Introduction Let L(group) be the first order language of group theory. One says that groups G and H are elementarily equivalent and writes G ≡ H if each sentence of L(group) which holds in one of these groups holds also the other one. There are many examples of pairs of elementarily equivalent groups which are not isomorphic. For example, the group Z is elementarily equivalent to every nonprincipal ultrapower of it although it is not isomorphic to it. Less trivial examples are given by the following result: If G and H are groups satisfying G× Z ∼= H × Z, then G ≡ H [Oge91] (see [Hir69] for an example of non-isomorphic groups G and H satisfying G × Z ∼= H × Z.) More generally, Nies points out in [Nie03, p. 288] that for every infinite finitely generated abstract group G there exists a countable group H such that G ≡ H but G 6∼= H. See also related results of Zil’ber in [Zil71] and Sabbagh and Wilson in [SaW91]. One of the consequences of the solution of Tarski’s problem is that all finitely generated free nonabelian groups are elementarily equivalent [Sel03, Thm. 3]. We refer the reader to [FrJ05, Chap. 7] for notions and results in logic and model theory that we use here. The goal of this note is to show that the situation is quite different in the category of profinite groups. Note that in this category “homomorphism” means “continuous homomorphism” and we say that a profinite group G is finitely generated if G has a dense finitely generated abstract subgroup; more generally we use the convention of [FrJ05, Chaps. 1, 17, and 22] for profinite groups. However, whenever we say that two profinite groups are elementarily equivalent, we mean that they are elementarily equivalent as abstract groups, i.e. in the sense defined in the preceding paragraph. Theorem A: Let G and H be elementarily equivalent profinite groups. If one of the groups is finitely generated, then they are isomorphic. The proof of Theorem A uses tools developed by Nikolov and Segal in their proof of the following deep result: Every abstract subgroup H of a finitely generated profinite group G with (G : H) < ∞ is open [NiS03 or NiS07]. Among others, that result relies on the classification of finite simple groups. Theorem A does not remain true if neither of the groups G and H is finitely 1 generated. An example to this situation appears in our second main result: Theorem B: Every two free pro-p Abelian groups of infinite rank are elementarily equivalent. The proof of Theorem B in Section 2 is based on the fact that every closed subgroup of a free Abelian pro-p group F is again a free Abelian pro-p group. An essential ingredient in the proof is a separation property saying that if rank(F ) = ∞ and x1, . . . , xn ∈ F , then F can be presented as a direct sum F = F0 ⊕ F1 such that rank(F0) = א0, rank(F1) ≥ א0, and x1, . . . , xn ∈ F0. The referee pointed out to us that Theorem B follows also from a deep result of Szmielew [Szm] that gives a general criterion for Abelian groups to be elementarily equivalent. This approach is explained in Section 3. We thank the referee for calling our attenion to the work of Smielew as well as for mentioning the results of Nies, Zil’ber, and Sabbagh-Wilson.