Abstract
Let G be a group. We denote by \({\nu(G)}\) an extension of the non-abelian tensor square \({G \otimes G}\) by \({G \times G}\). We prove that if G is finite-by-nilpotent, then the non-abelian tensor square \({G \otimes G}\) is finite-by-nilpotent. Moreover, \({\nu(G)}\) is nilpotent-by-finite (Theorem A). Also we characterize BFC-groups in terms of \({\nu(G)}\) among the groups G in which the derived subgroup is finitely generated (Theorem B).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Blyth R.D., Morse R.F.: Computing the nonabelian tensor square of polycyclic groups. J. Algebra 321, 2139–2148 (2009)
Brown R., Loday J.-L.: Van Kampen theorems for diagrams of spaces. Topology 26, 311–335 (1987)
Brown R., Johnson D.L., Robertson E.F.: Some computations of non-abelian tensor products of groups. J. Algebra 111, 177–202 (1987)
Gumber D., Kalra H.: On the converse of a theorem of Schur. Arch. Math. 101, 17–20 (2013)
Ph. Hall, Finite-by-nilpotent groups, Proc. Camb. Philos. Soc. 52 (1956), 611–616
Hilton P.: On a theorem of Schur. Int. J. Math. Math. Sci. 28, 455–460 (2001)
L.-C. Kappe, Nonabelian tensor products of groups: the commutator connection, Proc. Groups St. Andrews 1997 at Bath, London Math. Soc. Lecture Notes 261 (1999), 447–454.
B. C. R. Lima and R. N De Oliveira, Weak commutativity between two isomorophic polycyclic groups, J. Group. Theory 19 (2016), 239–248.
Moravec P.: The exponents of nonabelian tensor products of groups. J. Pure Appl. Algebra 212, 1840–1848 (2008)
Nakaoka I.N., Rocco N.R.: A survey of non-abelian tensor products of groups and related constructions. Bol. Soc. Paran. Mat. 30, 77–89 (2012)
Neumann B.H.: Groups with finite classes of conjugate elements. Prof. London Math. Soc. 1, 178–187 (1951)
Niroomand P.: The converse of Schur’s theorem. Arch. Math. 94, 401–403 (2010)
Niroomand P., Parvizi M.: On the structure of groups whose exterior or tensor square is a p-group. J. Algebra 352, 347–353 (2012)
D. J. S. Robinson, A course in the theory of groups, 2nd edition, Springer-Verlag, New York, 1996.
Rocco N.R.: On a construction related to the non-abelian tensor square of a group, Bol. Soc. Brasil Mat. 22, 63–79 (1991)
Rocco N.R.: A presentation for a crossed embedding of finite solvable groups. Comm. Algebra 22, 1975–1998 (1994)
M K. Yadav, Converse of Schur’s Theorem – A statement, preprint available at arXiv:1212.2710v2 [math.GR].
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by CAPES-Brazil.
Rights and permissions
About this article
Cite this article
Bastos, R., Rocco, N.R. Non-abelian tensor square of finite-by-nilpotent groups. Arch. Math. 107, 127–133 (2016). https://doi.org/10.1007/s00013-016-0930-2
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-016-0930-2