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Characteristic Subgroups of Free Groups

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Part of the Lecture Notes in Mathematics book series (LNM,volume 372)

Abstract

In [4], B.H. Neumann asked whether it is true that every characteristic subgroup W of a free group F of infinite rank is fully invariant, and in [51 he conjectured that this is so. Cohen [1] provided support for the conjecture by proving that W is always fully invariant when F/W is abelian-by-nilpotent. However two examples will be described here of characteristic subgroups of the free group F of countable rank which are not fully invariant. Also, a proof will be given of the fact that there are continuously many characteristic subgroups of F which are not fully invariant.

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References

  1. D.E. Cohen, “Characteristic subgroups of some relatively free groups”, J. London Math. Soc. 43 (1968), 445–451. MR37#1450.

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  4. B.H. Neumann, “Ascending verbal and Frattini series”, Math. Z. 69 (1958), 164–172. MR20#3218.

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  5. B.H. Neumann, “On characteristic subgroups of free groups”, Math. Z. 94 (1966), 143–151. MR35#240.

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  6. Hanna Neumann, Varieties of groups, (Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 37. Springer-Verlag, Berlin, Heidelberg, New York, 1967). MR35#6734.

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  7. M.F. Newman, “Just non-finitely-based varieties of groups”, Bull. Austral. Math. Soc. 4 (1971), 343–348. MR43#4891.

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  8. А.Ю. Ольшанский [A.Ju. Ol’šanskiĭ], “О характеристических подгруппах свободных групп” [On characteristic subgroups of free groups], Uspehi Mat. Nauk (to appear).

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© 1974 Springer-Verlag Berlin Heidelberg

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Bryant, R.M. (1974). Characteristic Subgroups of Free Groups. In: Newman, M.F. (eds) Proceedings of the Second International Conference on the Theory of Groups. Lecture Notes in Mathematics, vol 372. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-21571-5_11

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  • DOI: https://doi.org/10.1007/978-3-662-21571-5_11

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-06845-7

  • Online ISBN: 978-3-662-21571-5

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