## Illinois Journal of Mathematics

### 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.

#### Article information

Source
Illinois J. Math., Volume 47, Number 1-2 (2003), 223-236.

Dates
First available in Project Euclid: 17 November 2009

https://projecteuclid.org/euclid.ijm/1258488149

Digital Object Identifier
doi:10.1215/ijm/1258488149

Mathematical Reviews number (MathSciNet)
MR2031317

Zentralblatt MATH identifier
1041.20023

Subjects
Primary: 20F19: Generalizations of solvable and nilpotent groups
Secondary: 20E25: Local properties

#### Citation

Göbel, R.; Shelah, S.; Wallutis, S. L. On universal and epi-universal locally nilpotent groups. Illinois J. Math. 47 (2003), no. 1-2, 223--236. doi:10.1215/ijm/1258488149. https://projecteuclid.org/euclid.ijm/1258488149