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2011 | 9 | 3 | 616-626
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Almost fixed-point-free automorphisms of prime order

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Let ϕ be an automorphism of prime order p of the group G with C G(ϕ) finite of order n. We prove the following. If G is soluble of finite rank, then G has a nilpotent characteristic subgroup of finite index and class bounded in terms of p only. If G is a group with finite Hirsch number h, then G has a soluble characteristic subgroup of finite index in G with derived length bounded in terms of p and n only and a soluble characteristic subgroup of finite index in G whose index and derived length are bounded in terms of p, n and h only. Here a group has finite Hirsch number if it is poly (cyclic or locally finite). This is a stronger notion than that used in [Wehrfritz B.A.F., Almost fixed-point-free automorphisms of order 2, Rend. Circ. Mat. Palermo (in press)], where the case p = 2 is discussed.
Twórcy
Bibliografia
  • [1] Endimioni G., Polycyclic group admitting an almost regular automorphism of prime order, J. Algebra, 2010, 323(11), 3142–3146 http://dx.doi.org/10.1016/j.jalgebra.2010.03.015
  • [2] Fong P., On orders of finite groups and centralizers ofp-elements, OsakaJ.Math., 1976, 13(3), 483–489
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  • [4] Hartley B., Centralizers in locally finite groups, In: Group Theory, Brixen, 1986, Lecture Notes in Math., 1281, Springer, Berlin, 1987, 36–51 http://dx.doi.org/10.1007/BFb0078689
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  • [6] Kegel O.H., Wehrfritz B.A.F., Locally Finite Groups, North-Holland Math. Library, 3, North-Holland, Amsterdam-London, 1973
  • [7] Khukhro E.I., Nilpotent Groups and their Automorphisms, de Gruyter Exp. Math., 8, Walter de Gruyter, Berlin, 1993
  • [8] Khukhro E.I., p-Automorphisms of Finite p-Groups, London Math. Soc. Lecture Note Ser., 246, Cambridge Univ. Press, Cambridge, 1998 http://dx.doi.org/10.1017/CBO9780511526008
  • [9] Mal’tsev A.I., On certain classes of infinite soluble groups, Mat. Sb., 1951, 28(3), 567–588 (in Russian)
  • [10] Robinson D.J.S., Finiteness Conditions and Generalized Soluble Groups. I–II, Ergeb. Math. Grenzgeb., 62–63, Springer, Berlin-Heidelberg-New York, 1972
  • [11] Wehrfritz B.A.F., Infinite Linear Groups, Ergeb. Math. Grenzgeb., 76, Springer, Berlin-Heidelberg-New York, 1973
  • [12] Wehrfritz B.A.F., Groupand Ring Theoretic Properties of Polycyclic Groups, Algebr. Appl., 10, Springer, Dordrecht, 2009 http://dx.doi.org/10.1007/978-1-84882-941-1
  • [13] Wehrfritz B.A.F., Almost fixed-point-free automorphisms of soluble groups, J. Pure Appl. Algebra, 2011, 215(5), 1112–1115 http://dx.doi.org/10.1016/j.jpaa.2010.07.017
  • [14] Wehrfritz B.A.F., Almost fixed-point-free automorphisms of order 2, Rend. Circ. Mat. Palermo (in press)
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bwmeta1.element.doi-10_2478_s11533-011-0017-z
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