, Volume 51, Issue 4, pp 285-305

Properties and universal theories for partially commutative nilpotent metabelian groups

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Partially commutative nilpotent metabelian groups are considered. We describe how annihilators of elements of the commutator subgroup of a group G, as well as centralizers of elements of G in its commutator subgroup G′, are structured. It turns out that in the case where a defining graph of a group is a tree, the intersection of centralizers of distinct vertices and G′ coincides with the last nontrivial commutator subgroup of G. Universal theories for partially commutative nilpotent metabelian groups are compared: conditions on defining graphs of two partially commutative nilpotent metabelian groups are formulated which are sufficient for the two groups to have equal universal theories; conditions on defining graphs of two partially commutative metabelian groups are specified which are sufficient for the two groups to be universally equivalent; a criterion is given that decides whether two partially commutative nilpotent metabelian groups defined by trees are universally equivalent.

Translated from Algebra i Logika, Vol. 51, No. 4, pp. 429-457, July-August, 2012.