On universal and epi-universal locally nilpotent groups

Abstract

In this paper we are mainly concerned with the class $\mathcal{LN}$ of all locally nilpotent groups. Using similar arguments as in [grs] we first show that there is no universal group in $\mathcal{LN}_\lambda$ if $\lambda$ is a cardinal such that $\lambda=\lambda^{\aleph_0}$; here we call a group $G$ universal (in $\mathcal{LN}_\lambda$) if any group $H\in\mathcal{LN}_\lambda$ can be embedded into $G$, where $\mathcal{LN}_\lambda$ denotes the class of all locally nilpotent groups of cardinality at most $\lambda$. However, our main interest is in the construction of torsion-free epi-universal groups in $\mathcal{LN}_\lambda$, where $G\in\mathcal{LN}_\lambda$ is said to be epi-universal if any group $H\in\mathcal{LN}_\lambda$ 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 $\mathcal{LN}$ to other canonical classes of groups to construct epi-universal objects.