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Spring/Summer 2003 On universal and epi-universal locally nilpotent groups
R. Göbel, S. Shelah, S. L. Wallutis
Illinois J. Math. 47(1-2): 223-236 (Spring/Summer 2003). DOI: 10.1215/ijm/1258488149


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.


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R. Göbel. S. Shelah. S. L. Wallutis. "On universal and epi-universal locally nilpotent groups." Illinois J. Math. 47 (1-2) 223 - 236, Spring/Summer 2003.


Published: Spring/Summer 2003
First available in Project Euclid: 17 November 2009

zbMATH: 1041.20023
MathSciNet: MR2031317
Digital Object Identifier: 10.1215/ijm/1258488149

Primary: 20F19
Secondary: 20E25

Rights: Copyright © 2003 University of Illinois at Urbana-Champaign


Vol.47 • No. 1-2 • Spring/Summer 2003
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