Stability of nilpotent groups of class 2 and prime exponent

Abstract

Let p be an odd prime. A method is described which given a structure M of finite similarity type produces a nilpotent group of class 2 and exponent p which is in the same stability class as M.Theorem. There are nilpotent groups of class 2 and exponent p in all stability classes.Theorem. The problem of characterizing a stability class is equivalent to characterizing the (nilpotent, class 2, exponent p) groups in that class.

The classification problem for von Neumann factors

“…By [15,Lemma 2.2] it follows that Γ F 2 Γ F 2 . Since the connected components of Γ F 2 and Γ F 2 are all isomorphic to Γ and Γ , respectively, it follows that Γ Γ .…”

Stable pseudofinite groups

“…We construct an ω-stable pseudofinite group which is not abelian-by-finite. The construction uses Mekler's method for building nilpotent class 2 groups from graphs, preserving various model-theoretic properties [8]. We follow the explanation in [6, Appendix A.3], using the notion of special model also described in [6].…”

“…It follows that G(Δ(λ)) is ω-stable. (For this, see either [8,Theorem 2.11], or [6, Corollary A.3.19], whose proof does not use uncountability of λ.) Furthermore, if H is a special model of cardinality λ of Th(G(Δ(λ))), then H ∼ = G(Δ(λ) + ) × V (λ), by A.3.15 of [6].…”

Isomorphisms, definable relations, and Scott families of class 2 nilpotent groups

“…x i is an Abelian p-group, generated by the In [4], a method of coding a symmetric irreflexive graph into a metabelian group was proposed for studying model-theoretic properties of the group. We will use this coding in order to prove Prop.…”