The Relatively Free Groups Satisfy Noncentral Commutative Transitivity

We prove that a free group, , relative to the variety, , of all groups simultaneously nilpotent of class at most and metabelian is such that the centralizer of every noncentral element is abelian. We relate that result to the model theory of such groups as well as a quest to find a relative analog in of a classical theorem of Benjamin Baumslag. We also touch briefly on similar considerations in the varieties of nilpotent groups. 1. Introduction 1.1. Separation and Discrimination Definition 1. Let be a group and a nonempty class of groups closed under isomorphism. separates provided to every nontrivial element ; there is a group and a homomorphism such that . If is a positive integer, then -separates provided to every ; there is a group and a homomorphism such that for all . discriminates provided -separates for every positive integer . Note that separation is the same as -separation. In the event that is the isomorphism class of a single group we may say that separates (-separates, discriminates) for separates (-separates, discriminates) . Observe that separating is equivalent to being embeddable in a direct product of groups in . In particular, this is so if is isomorphic to a subdirect product of groups in (see [1]). Here is a subdirect product of the family provided that, for each fixed , the restriction of the projection onto the th coordinate remains surjective. Definition 2. Let be a nonempty class of groups closed under isomorphism. is a prevariety if it is closed under taking subgroups and direct products. B. Fine observed that is a prevariety if and only if it is a separating family in the sense that every group separated by must already lie in . Definition 3. Let be a nonempty class of groups closed under isomorphism. For a given group let be the set . is a radical class provided, for every group , If is a radical class, then is the radical of relative to . We observe that every prevariety is a radical class with Here . Let us call the isomorphism class of the one element group trivial. Every nontrivial prevariety admits free objects of every rank. Indeed, if is a free group of rank , then is free of rank relative to . Definition 4. One will say a group is freely separated (freely discriminated) provided it is separated (discriminated) by the class of free groups. Definition 5. A group is commutative transitive or CT provided the binary relation , given by if and only if , is transitive. Remark 6. Clearly is CT if and only if the centralizer in of any nontrivial element is abelian. This result was originally stated by Harrison [2]. Theorem 7