The nilpotent genus of finitely generated residually nilpotent groups
-
Niamh O’Sullivan
Abstract
Let 𝐺 and 𝐻 be residually nilpotent groups. Then 𝐺 and 𝐻 are in the same nilpotent genus if they have the same lower central quotients (up to isomorphism). A potentially stronger condition is that 𝐻 is para-𝐺 if there exists a monomorphism of 𝐺 into 𝐻 which induces isomorphisms between the corresponding quotients of their lower central series. We first consider finitely generated residually nilpotent groups and find sufficient conditions on the monomorphism so that 𝐻 is para-𝐺. We then prove that, for certain polycyclic groups, if 𝐻 is para-𝐺, then 𝐺 and 𝐻 have the same Hirsch length. We also prove that the pro-nilpotent completions of these polycyclic groups are locally polycyclic.
Acknowledgements
I would like to thank the anonymous referee for helpful comments and suggestions.
-
Communicated by: Benjamin Klopsch
References
[1] S. Andreadakis, On the automorphisms of free groups and free nilpotent groups, Proc. Lond. Math. Soc. (3) 15 (1965), 239–268. 10.1112/plms/s3-15.1.239Search in Google Scholar
[2] G. Baumslag, Parafree groups, Infinite Groups: Geometric, Combinatorial and Dynamical Aspects, Progr. Math. 248, Birkhäuser, Basel (2005), 1–14. 10.1007/3-7643-7447-0_1Search in Google Scholar
[3] G. Baumslag, R. Mikhailov and K. E. Orr, A new look at finitely generated metabelian groups, Computational and Combinatorial Group Theory and Cryptography, Contemp. Math. 582, American Mathematical Society, Providence (2012), 21–37. 10.1090/conm/582/11551Search in Google Scholar
[4] G. Baumslag, R. Mikhailov and K. E. Orr, Localization, metabelian groups, and the isomorphism problem, Trans. Amer. Math. Soc. 369 (2017), no. 10, 6823–6852. 10.1090/tran/6838Search in Google Scholar
[5] M. R. Bridson and A. W. Reid, Nilpotent completions of groups, Grothendieck pairs, and four problems of Baumslag, Int. Math. Res. Not. IMRN 2015 (2015), no. 8, 2111–2140. 10.1093/imrn/rnt353Search in Google Scholar
[6] M. Hall, Jr., The Theory of Groups, The Macmillan, New York, 1969. Search in Google Scholar
[7] J. C. Lennox and D. J. S. Robinson, The Theory of Infinite Soluble Groups, Oxford Math. Monogr., The Clarendon, Oxford, 2004. 10.1093/acprof:oso/9780198507284.001.0001Search in Google Scholar
[8] A. I. Mal’cev, On some classes of infinite soluble groups, Mat. Sbornik N. S. 28(70) (1951), 567–588. Search in Google Scholar
[9] A. L. Mal’cev, On certain classes of infinite solvable groups, Amer. Math. Soc. Transl. (2) 2 (1956), 1–21. 10.1090/trans2/002/01Search in Google Scholar
[10] D. J. S. Robinson, A Course in the Theory of Groups, Grad. Texts in Math. 80, Springer, New York, 1996. 10.1007/978-1-4419-8594-1Search in Google Scholar
[11] D. J. S. Robinson, A. Russo and G. Vincenzi, On groups which contain no HNN-extensions, Internat. J. Algebra Comput. 17 (2007), no. 7, 1377–1387. 10.1142/S0218196707004311Search in Google Scholar
[12] D. Segal, Polycyclic Groups, Cambridge Tracts in Math. 82, Cambridge University, Cambridge, 1983. Search in Google Scholar
[13] A. L. Šmel’kin, Polycyclic groups, Sibirsk. Mat. Ž. 9 (1968), 234–235. Search in Google Scholar
© 2023 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Elementary amenable groups of cohomological dimension 3
- The nilpotent genus of finitely generated residually nilpotent groups
- Quantifying lawlessness in finitely generated groups
- On multidimensional Schur rings of finite groups
- A classification of the prime graphs of pseudo-solvable groups
- On generations by conjugate elements in almost simple groups with socle 2𝐹4(𝑞2)′
- A characterization of the simple Ree groups 2𝐹4(𝑞2) by their character codegrees
- A characterization of finite groups having a single Galois conjugacy class of certain irreducible characters
- Class-two quotients of finite permutation groups
- A note on 𝑑-maximal 𝑝-groups
Articles in the same Issue
- Frontmatter
- Elementary amenable groups of cohomological dimension 3
- The nilpotent genus of finitely generated residually nilpotent groups
- Quantifying lawlessness in finitely generated groups
- On multidimensional Schur rings of finite groups
- A classification of the prime graphs of pseudo-solvable groups
- On generations by conjugate elements in almost simple groups with socle 2𝐹4(𝑞2)′
- A characterization of the simple Ree groups 2𝐹4(𝑞2) by their character codegrees
- A characterization of finite groups having a single Galois conjugacy class of certain irreducible characters
- Class-two quotients of finite permutation groups
- A note on 𝑑-maximal 𝑝-groups
