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Preservation of saturation and stability in a variety of nilpotent groups

This paper is a contribution to the growing literature on the model theory of nilpotent groups. (See Baumslag and Levin [2]; Eršov [5]; Hodges [9], [10]; Mal′cev [14]; Olin [16] and Saracino [19], [20].) In it we investigate the conditions under which the free product in the variety of all nilpotent of class 2 (nil-2) groups preserves saturation and stability. It is well known that the direct product preserves both saturation (see Waszkiewicz and Wȩglorz [23]) and stability (see Wierzejewski [24]; Macintyre [13]; Eklof and Fisher [4]). On the other hand it is easy to show that the full free product of groups preserves neither property; indeed, in the case of saturation this failure is extremely bad since no free product of nontrivial groups is even 2-saturated. Our results show that the nil-2 free product falls between these two extremes. Our proofs are mainly model-theoretic with a smattering of elementary algebra and rely heavily upon the unique normal form for the elements of a nil-2 free product given by MacHenry in [12]. (This normal form and some of its consequences are discussed in §1.) We assume familiarity with the basic ideas of saturation (see Chapter 5 of [3]) and Shelah's treatment of stability in [22]. We prove two main theorems in §3 each giving a necessary and sufficient condition in separate situations for the preservation of saturation. In the first (Theorem 3.1) we allow one finite factor, while in the second (Theorem 3.10) we deal solely with torsion groups. Our motivation for the proof of sufficiency was the paper of Waszkiewicz and Wȩglorz [23] and the principal tool is a “Feferman-Vaught” Theorem for the nil-2 free product which we prove in §2. We also show that if both factors in a nil-2 free product are nontorsion and one factor has a nil-2 basis, then the group is not even 3-saturated. We leave open the case where both factors are infinite but only one is torsion.