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.
@article{bwmeta1.element.doi-10_2478_s11533-011-0017-z,
author = {Bertram Wehrfritz},
title = {Almost fixed-point-free automorphisms of prime order},
journal = {Open Mathematics},
volume = {9},
year = {2011},
pages = {616-626},
zbl = {1245.20037},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0017-z}
}
Bertram Wehrfritz. Almost fixed-point-free automorphisms of prime order. Open Mathematics, Tome 9 (2011) pp. 616-626. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0017-z/
[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 | Zbl 1202.20037
[2] Fong P., On orders of finite groups and centralizers ofp-elements, OsakaJ.Math., 1976, 13(3), 483–489 | Zbl 0372.20010
[3] Gorenstein D., Finite Groups, Harper’s Seriesin Modern Mathematics, Harper & Row, New York-Evanston-London, 1968 | Zbl 0185.05701
[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
[5] Huppert B., Blackburn N., Finite Groups II, Grundlehren Math. Wiss., 242, Springer, Berlin-Heidelberg-NewYork, 1982 | Zbl 0477.20001
[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 | Zbl 0897.20018
[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 | Zbl 1215.20030
[14] Wehrfritz B.A.F., Almost fixed-point-free automorphisms of order 2, Rend. Circ. Mat. Palermo (in press)