Finite axiomatizability for profinite groups

Andre Nies,Katrin Tent,Dan Segal

doi:10.1112/plms.12420

Abstract

A group is finitely axiomatizable (FA) in a class C if it can be determined up to isomorphism within C by a sentence in the first-order language of group theory. We show that profinite groups of various kinds are FA in the class of profinite groups, or in the class of pro- p groups for some prime p. Both algebraic and model-theoretic methods are developed for the purpose. Reasons why certain groups cannot be FA are also discussed.

Highlights

A group is finitely axiomatizable (FA) in a class C if it can be determined up to isomorphism within C by a sentence in the first-order language of group theory
We consider the question: which profinite groups can be characterized by a single sentence? To make this more precise, let us say that a group G is FA in C if C is a class of groups containing G, L is a language, and there is a sentence σG of L such that for any group H in C, H |= σG if and only if H ∼= G
When proving that a certain group G is FA in some class C, we often establish a stronger property, namely: for some finite tuple g in G, there is a formula σG such that for a group H in C and a tuple h in H, H |= σG(h) if and only if there is an isomorphism from G to H mapping g to h, a situation denoted by (G, g) ∼= (H, h)

Summary

Definable subgroups

Xk) has width f and Gw = {w(g)±1 | g ∈ G(k)} This formula defines a closed subset in every profinite group, since the verbal mapping G(k) → G defined by w is continuous, has compact image. Note that subgroups like w(G) when w is a word of finite width are definable as in (1) without parameters. When proving that a certain group G is FA in some class C, we often establish a stronger property, namely: for some finite (usually generating) tuple g in G, there is a formula σG such that for a group H in C and a tuple h in H, H |= σG(h) if and only if there is an isomorphism from G to H mapping g to h, a situation denoted by (G, g) ∼= (H, h) In this case we say that (G, g) is FA in C. This implies that G is FA in C: for H in C, we have H ∼= G if and only if H |= ∃x.σG(x)

Finite extensions

Bi-interpretation

Conclusion

Profinite groups of finite rank

Special linear groups

Some negative results