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Automorphisms of free nilpotent-by-abelian groups

Published online by Cambridge University Press:  24 October 2008

R. M. Bryant  and
C. K. Gupta
R. M. Bryant
Affiliation:
University of Manchester Institute of Science and Technology, Manchester M60 1QD
C. K. Gupta
Affiliation:
University of Manitoba, Winnipeg R3T 2N2, Canada
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Extract

Let Fn be a free group of finite rank n with basis {x1,…, xn}. Let be a variety of groups and write for the verbal subgroup of Fn corresponding to . (See [11] for information on varieties and related concepts.) Every automorphism of Fn induces an automorphism of the relatively free group Fn/V, and those automorphisms of Fn/V arising in this way are called tame. If is the variety of all metabelian groups and n ╪ 3 then every automorphism of Fn/V is tame [2, 4, 12]. But this is an exceptional situation. For many (and probably most) other varieties , Fn/V has non-tame automorphisms for all sufficiently large n. This holds for the variety of all nilpotent groups of class at most c where c ≥ 3 [1, 3] and for nearly all product varieties including, in particular, the variety of all groups whose derived groups are nilpotent of class at most c, where c > 2 [10, 13].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 114 , Issue 1 , July 1993 , pp. 143 - 147
Copyright
Copyright © Cambridge Philosophical Society 1993

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References

REFERENCES

[1]Andreadakis, S.. On the automorphisms of free groups and free nilpotent groups. Proc. London Math. Soc. (3) 15 (1965), 239–268.CrossRefGoogle Scholar
[2]Bachmuth, S.. Automorphisms of free metabelian groups. Trans. Amer.Maih. Soc. 118 (1965), 93–104.CrossRefGoogle Scholar
[3]Bachmuth, S.. Induced automorphisms of free groups and free metabelian groups. Trans. Amer. Math. Soc. 122 (1966), 1–17.CrossRefGoogle Scholar
[4]Bachmuth, S. and Mochizuki, H. Y..Aut(F)→Aut(F/Fn) is surjective for free group F of rank ≥ 4. Trans. Amer. Math. Soc. 292 (1985), 81–101.Google Scholar
[5]Bryant, R. M. and Geoves, J. R. J.. On automorphisms of relatively free groups. J. Algebra 137 (1991), 195–205.CrossRefGoogle Scholar
[6]Bryant, R. M. and Groves, J. R. J.. Automorphisms of free metabelian groups of infinite rank. Comm. Algebra 20 (1992), 783–814.CrossRefGoogle Scholar
[7]Bryant, R. M. and Macedonska, O.. Automorphisms of relatively free nilpotent groups of infinite rank. J. Algebra 121 (1989), 388–398.CrossRefGoogle Scholar
[8]Gupta, C. K. and Gupta, N. D.. Lifting primitivity of free nilpotent groups. Proc. Amer. Math. Soc. 114 (1992), 617–621.CrossRefGoogle Scholar
[9]Gupta, C. K., Gupta, N. D. and Roman'kov, V. A.. Primitivity in free groups and free metabelian groups. Canad. J. Math. 44 (1992), 516–523.CrossRefGoogle Scholar
[10]Gupta, C. K. and Levin, F.. Tame range of automorphism groups of free polynilpotent groups. Comm. Algebra 19 (1991), 2497–2500.CrossRefGoogle Scholar
[11]Neumann, H.. Varieties of groups (Springer, Berlin, 1967).CrossRefGoogle Scholar
[12]Roman'kov, V. A.. Automorphism groups of free metabelian groups. Problems of Relationships of Abstract and Applied Algebra (Novosibirsk, 1985), 53–80.Google Scholar
[13]Shpilrain, V. E.. Automorphisms of F/R' groups. Internat. J. Algebra and Comput. 1 (1991), 177–184.CrossRefGoogle Scholar