Table of Contents
Volume 2014, Article ID 379030, 12 pages
Research Article

The Relatively Free Groups Satisfy Noncentral Commutative Transitivity

1Department of Mathematics, U.S. Naval Academy, Annapolis, MD 21402, USA
2Department of Mathematics, Temple University, Philadelphia, PA 19122, USA
3Department of Statistics, Temple University, Philadelphia, PA 19122, USA

Received 22 May 2014; Accepted 25 July 2014; Published 1 October 2014

Academic Editor: Antonio M. Cegarra

Copyright © 2014 Anthony M. Gaglione et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.