Normal automorphisms of the metabelian product of free abelian Lie algebras

Let M be the metabelian product of free abelian Lie algebras of finite rank. In this study we prove that every normal automorphism of M is an IA-automorphism and acts identically on M′.

On 2-Symmetric Words and Verbal Subgroup of Metabelian Product of Abelian Groups

Let F be the free group of rank 2 with basis {x, y}, and G a metabelian product of some non-trivial abelian groups. If not all the factors of G are torsion groups, it is proved that the verbal subgroup of G in F equals F″. Moreover, all the 2-symmetric words of G are determined by using left Fox derivatives. In addition, we provide an example to illustrate that if all the factors of G are torsion groups, the above results need not be true.