Mekler's construction preserves [formula omitted]-triviality

Abstract

For every structure M of finite signature Mekler (J. Symbolic Logic 46 (1981) 781) has constructed a group G such that for every κ the maximal number of n-types over an elementary equivalent model of cardinality κ is the same for M and G. These groups are nilpotent of class 2 and of exponent p, where p is a fixed prime greater than 2. We consider stable structures M only and show that M is CM-trivial if and only if G is CM-trivial. Furthermore, we obtain that the free group F 2(p,ω) in the variety of 2-nilpotent groups of exponent p>2 with ω free generators has a CM-trivial ω-stable theory.