On universal and EPI-universal locally nilpotent groups

Abstract

In this paper we are mainly concerned with the class ℒscript N sign of all locally nilpotent groups. Using similar arguments as in [2] we first show that there is no universal group in ℒscript N signλ if λ is a cardinal such that λ = λא0; here we call a group G universal (in ℒscript N signλ) if any group H ∈ ℒscript N signλ can be embedded into G, where ℒscript N signλ denotes the class of all locally nilpotent groups of cardinality at most λ. However, our main interest is in the construction of torsion-free epi-universal groups in ℒscript N signλ, where G ∈ ℒscript N sign λ is said to be epi- universal if any group H ∈ ℒscript N signλ is an epimorphic image of G. Thus we give an affirmative answer to a question of Plotkin. To prove the torsion-freeness of the constructed locally nilpotent group we adjust the well-known commutator collecting process due to P. Hall to our situation. Finally, we briefly discuss how to apply the methods we used for the class ℒscript N sign to other canonical classes of groups to construct epi-universal objects.