### On universal and epi-universal locally nilpotent groups

by Goebel and Shelah and Wallutis. [GShW:742]

Illinois J Math, 2003

In this paper we mainly consider the class LN of all locally
nilpotent groups. We first show that there is no universal group
in
LN_lambda if lambda is a cardinal such that
lambda = lambda^{aleph_0} ; here we call a group G universal
(in LN_lambda) if any group H in LN_lambda can be
embedded into G where LN_lambda denotes the class of all
locally nilpotent groups of cardinality at most lambda . However,
our main interest is the construction of torsion-free epi-universal
groups in LN_lambda, where G in LN_lambda is said to be epi-universal
if any group H in LN_lambda is an epimorphic image
of G . Thus we give an affirmative answer to a question by
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 use the same methods as for the class LN for other
canonical classes of groups to construct epi-universal objects.

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