Hostname: page-component-848d4c4894-wzw2p Total loading time: 0 Render date: 2024-05-20T18:35:54.242Z Has data issue: false hasContentIssue false
  • English
  • Français

Infinitely generated free nilpotent groups: completeness of the automorphism groups

Published online by Cambridge University Press:  13 July 2009

VLADIMIR TOLSTYKH
VLADIMIR TOLSTYKH*
Affiliation:
Department of Mathematics, Yeditepe University, 34755 Kayɩşdağɩ, Istanbul, Turkey. e-mail: tvlaa@rambler.ru
Get access Rights & Permissions [Opens in a new window]

Abstract

We transfer the results of Dyer, Formanek and Kassabov on the automorphism towers of finitely generated free nilpotent groups to infinitely generated free nilpotent groups. We prove that the automorphism groups of infinitely generated free nilpotent groups are complete. By combining the results of Dyer, Formanek and Kassabov with the results in this paper, one gets that the automorphism tower of any free nilpotent group terminates after finitely many steps.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 147 , Issue 3 , November 2009 , pp. 541 - 566
Copyright
Copyright © Cambridge Philosophical Society 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Andreadakis, S.On the automorphisms of free groups and free nilpotent groups. Proc. London Math. Soc 15 (1965), 239–268.CrossRefGoogle Scholar
[2]Bahturin, Yu.Basic structures of modern algebra. Mathematics and its Applications, 265 (Kluwer Academic Publishers Group, 1993).CrossRefGoogle Scholar
[3]Bridson, M. and Vogtmann, K.Automorphisms of automorphism groups of free groups. J. Algebra 229 (2000), 785–792.CrossRefGoogle Scholar
[4]Bryant, R. M. and Gupta, C. K.Automorphism groups of free nilpotent groups. Arch. Math. (Basel) 52 (1989), 313–320.CrossRefGoogle Scholar
[5]Bryant, R. M. and Roman'kov, V. A.The automorphism groups of relatively free algebras. J. Algebra 209 (1998), 713–723.CrossRefGoogle Scholar
[6]Burns, R. G. and Pi, L.Generators for the bounded automorphisms of infinite-rank free nilpotent groups. Bull. Austral. Math. Soc. 40 (1989), 175–187.CrossRefGoogle Scholar
[7]Dixon, J. and Mortimer, B.Permutation groups. Graduate Texts in Mathematics, 163 (Springer-Verlag, 1996).CrossRefGoogle Scholar
[8]Dyer, J. and Formanek, E.The automorphism group of a free group is complete. J. London Math. Soc. 11 (1975), 181–190.CrossRefGoogle Scholar
[9]Dyer, J. and Formanek, E.Automorphism sequences of free nilpotent group of class two. Math. Proc. Camb. Phil. Soc. 79 (1976), 271–279.CrossRefGoogle Scholar
[10]Dyer, J. and Formanek, E.Characteristic subgroups and complete automorphism groups. Amer. J. Math. 99 (1977), 713–753.CrossRefGoogle Scholar
[11]Formanek, E.Characterizing a free group in its automorphism group. J. Algebra. 133 (1990), 424–432.CrossRefGoogle Scholar
[12]Hodges, W.Model Theory (Cambridge University Press, 1993).CrossRefGoogle Scholar
[13]Kassabov, M. On the automorphism tower of free nilpotent groups. PhD. thesis. Yale University (2003).Google Scholar
[14]Khramtsov, D. G. Completeness of groups of outer automorphisms of free groups. Group-theoretic investigations (Akad. Nauk SSSR Ural. Otdel., Sverdlovsk, 1990), pp. 128–143.Google Scholar
[15]Magnus, W., Karrass, A. and Solitar, D.Combinatorial Group Theory (Wiley, 1966).Google Scholar
[16]Neumann, H.Varieties of Groups (Springer-Verlag, 1967).CrossRefGoogle Scholar
[17]Tolstykh, V.The automorphism tower of a free group. J. London Math. Soc. (2) 61 (2000), 423–440.CrossRefGoogle Scholar
[18]Tolstykh, V.Free two-step nilpotent groups whose automorphism group is complete. Math. Proc. Camb. Phil. Soc. 131 (2001), 73–90.CrossRefGoogle Scholar
[19]Tolstykh, V. On the logical strength of the automorphism groups of free nilpotent groups. Contemp. Math. 302 (AMS, Providence, 2002), pp. 113–120.Google Scholar
[20]Tolstykh, V.On the Bergman property for the automorphism groups of relatively free groups. J. London Math. Soc. (2) 73 (2006), 669–680.CrossRefGoogle Scholar