Finite groups in which the centralizer of every noncentral element of odd order is abelian

In this paper, we consider the influence of the number of centralizers of noncentral elements of odd order on the solvability of finite groups. Also, we characterize some finite groups in which the centralizer of every noncentral element of odd order is abelian.

The Relatively Free Groups F(Nc∧A2) Satisfy Noncentral Commutative Transitivity

We prove that a free group, F(Nc∧A2), relative to the variety, Nc∧A2, of all groups simultaneously nilpotent of class at most c 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 Nc∧A2 of a classical theorem of Benjamin Baumslag. We also touch briefly on similar considerations in the varieties Nc of nilpotent groups.