# Sh:742

- Göbel, R., Shelah, S., & Wallutis, S. L. (2003).
*On universal and epi-universal locally nilpotent groups*. Illinois J. Math.,**47**(1-2), 223–236. arXiv: math/0112252 http://projecteuclid.org/euclid.ijm/1258488149 MR: 2031317 -
Abstract:

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. - No downloadable versions available.

Bib entry

@article{Sh:742, author = {G{\"o}bel, R{\"u}diger and Shelah, Saharon and Wallutis, Simone L.}, title = {{On universal and epi-universal locally nilpotent groups}}, journal = {Illinois J. Math.}, fjournal = {Illinois Journal of Mathematics}, volume = {47}, number = {1-2}, year = {2003}, pages = {223--236}, issn = {0019-2082}, mrclass = {20F19 (20E25)}, mrnumber = {2031317}, mrreviewer = {Howard Smith}, url = {http://projecteuclid.org/euclid.ijm/1258488149}, note = {\href{https://arxiv.org/abs/math/0112252}{arXiv: math/0112252}}, arxiv_number = {math/0112252}, specialissue = {Special issue in honor of Reinhold Baer (1902--1979)} }