On groups admitting a~group of automorphisms whose centralizer has bounded rank

V. D. Mazurov ; E. I. Khukhro

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 3 (2006), pp. 257-283.

Résumé

We obtain restrictions on the structure of a finite group $G$ with a group of automorphisms $A$ in terms of the order of $A$ and the rank of the fixed-point subgroup $C_G(A)$. When $A$ is regular, that is, $C_G(A)=1$, there are well-known results giving in many cases the solubility of $G$, or bounds for the Fitting height. Some earlier “almost regular” results were deriving the solubility, or bounds for the Fitting height, of a subgroup of index bounded in terms of $|A|$ and $|C_G(A)|$. Now we prove rank analogues of these results: when “almost regular” in the hypothesis is interpreted as a restriction on the rank of $C_G(A)$, it is natural to seek solubility, or nilpotency, or bounds for the Fitting height, of “almost” entire group modulo certain bits of bounded rank. The classification is used to prove almost solubility. For soluble groups the Hall–Higman-type theorems are combined with the theory of powerful $p$-groups to obtain almost nilpotency, or bounds for the Fitting height of a normal subgroup with quotient of bounded rank. Examples are produced showing that some of our results are in a sense best-possible, while certain results on almost regular automorphism have no valid rank analogues. Several open problems are discussed, especially in the case of nilpotent $G$.

Bibliographie

[1] Yu. M. Gorchakov, “O suschestvovanii abelevykh podgrupp beskonechnykh rangov v lokalno razreshimykh gruppakh”, Dokl. AN SSSR, 146 (1964), 14–16

[2] V. A. Kreknin, “Razreshimost algebr Li s regulyarnym avtomorfizmom konechnogo poryadka”, Dokl. AN SSSR, 150 (1963), 467–469 | MR | Zbl

[3] V. A. Kreknin, A. I. Kostrikin, “Algebry Li s regulyarnym avtomorfizmom”, Dokl. AN SSSR, 149 (1963), 249–251 | MR | Zbl

[4] N. Yu. Makarenko, “Nilpotentnyi ideal v koltse Li s pochti regulyarnym avtomorfizmom prostogo poryadka”, Sibirsk. matem. zhurn., 46:6 (2005), 1361–1374 | MR

[5] Yu. I. Merzlyakov, “O lokalno razreshimykh gruppakh konechnogo ranga”, Algebra i logika, 3:2 (1964), 5–16 | MR

[6] E. I. Khukhro, “Konechnye $p$-gruppy, dopuskayuschie avtomorfizm poryadka $p$ s malym chislom nepodvizhnykh tochek”, Matem. zametki, 38 (1985), 652–657 | MR | Zbl

[7] E. I. Khukhro, “Gruppy i koltsa Li, dopuskayuschie pochti regulyarnyi avtomorfizm prostogo poryadka”, Matem. sbornik, 181 (1990), 1197–1219

[8] E. I. Khukhro, “Konechnye razreshimye i nilpotentnye gruppy s ogranicheniem na rang tsentralizatora avtomorfizma prostogo poryadka”, Sibirsk. matem. zhurn., 41 (2000), 451–469 | MR | Zbl

[9] E. I. Khukhro, V. D. Mazurov, “Gruppy s avtomorfizmom prostogo poryadka, chei tsentralizator imeet malyi rang”, Dokl. AN SSSR, 402 (2005), 740–742 | MR

[10] S. D. Bell and B. Hartley, “A note on fixed-point-free actions of finite groups”, Quart. J. Math. Oxford Ser. (2), 41:162 (1990), 127–130 | DOI | MR | Zbl

[11] T. R. Berger, “Nilpotent fixed point free automorphism groups of solvable groups”, Math. Z., 131 (1973), 305–312 | DOI | MR | Zbl

[12] R. Brauer and K. A. Fowler, “On groups of even order”, Ann. Math. (2), 62 (1955), 565–583 | DOI | MR | Zbl

[13] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of finite groups, Clarendon Press, Oxford, 1985 | MR | Zbl

[14] E. C. Dade, “Carter subgroups and Fitting heights of finite solvable groups”, Illinois J. Math., 13 (1969), 449–514 | MR | Zbl

[15] W. Feit and J. G. Thompson, “Solvability of groups of odd order”, Pacific J. Math., 13 (1963), 775–1029 | MR | Zbl

[16] P. Fong, “On orders of finite groups and centralizers of $ p$-elements”, Osaka J. Math., 13 (1976), 483–489 | MR | Zbl

[17] D. Gorenstein, Finite Groups, Chelsea, New York, 1968 | MR

[18] D. Gorenstein, Finite simple groups. An introduction to their classification, Plenum Publishing Corp., New York, USA, 1982 | MR | Zbl

[19] F. Gross, “Solvable groups admitting a fixed-point-free automorphism of prime power order”, Proc. Amer. Math. Soc., 17 (1966), 1440–1446 | DOI | MR | Zbl

[20] P. Hall and G. Higman, “On the $p$-length of $p$-soluble groups and reduction theorems for Burnside's problem”, Proc. London Math. Soc. (3), 6 (1956), 1–42 | DOI | MR | Zbl

[21] B. Hartley and T. Meixner, “Finite soluble groups containing an element of prime order whose centralizer is small”, Arch. Math. (Basel), 36 (1981), 211–213 | MR | Zbl

[22] G. Higman, “Groups and rings which have automorphisms without non-trivial fixed elements”, J. London Math. Soc. (2), 32 (1957), 321–334 | DOI | MR | Zbl

[23] B. Hartley and I. M. Isaacs, “On characters and fixed points of coprime operator groups”, J. Algebra, 131 (1990), 342–358 | DOI | MR | Zbl

[24] O. H. Kegel and B. A. F. Wehrfritz, Locally finite grous, North-Holland, Amsterdam; American Elsevier, New York, 1973 | MR | Zbl

[25] E. I. Khukhro and V. D. Mazurov, “Finite groups with an automorphism of prime order whose centralizer has small rank”, J. Algebra, 301 (2006), 474–492 | DOI | MR | Zbl

[26] E. I. Khukhro and V. D. Mazurov,, Proc. Groups St. Andrews'05, Cambridge Univ. Press, Cambridge, 2006 | MR

[27] L. G. Kovács, “On finite soluble groups”, Math. Z., 103 (1968), 37–39 | DOI | MR | Zbl

[28] H. Kurzweil, “$p$-Automorphismen von auflösbaren $p'$-Gruppen”, Math. Z., 120 (1971), 326–354 | DOI | MR | Zbl

[29] P. Longobardi and M. Maj, “On the number of generators of a finite group”, Arch. Math. (Basel), 50 (1988), 110–112 | MR | Zbl

[30] A. Lubotzky and A. Mann, “Powerful $p$-groups. I: Finite groups”, J. Algebra, 105 (1987), 484–505 | DOI | MR | Zbl

[31] N. Yu. Makarenko, Finite metabelian groups admitting an automorphism of prime order with a restriction on the rank of its centralizer, Written communication, 1999

[32] N. Yu. Makarenko and E. I. Khukhro, “Almost solubility of Lie algebras with almost regular automorphisms”, J. Algebra, 277 (2004), 370–407 | DOI | MR | Zbl

[33] M. R. Pettet, “Automorphisms and Fitting factors of finite groups”, J. Algebra, 72 (1981), 404–412 | DOI | MR | Zbl

[34] J. E. Roseblade, “On groups in which every subgroup is subnormal”, J. Algebra, 2 (1965), 402–412 | DOI | MR | Zbl

[35] E. E. Shult, “On groups admitting fixed point free abelian operator groups”, Illinois J. Math., 9 (1965), 701–720 | MR | Zbl

[36] P. Shumyatsky, “Involutory automorphisms of finite groups and their centralizers”, Arch. Math. (Basel), 71 (1998), 425–432 | MR | Zbl

[37] J. Thompson, “Finite groups with fixed-point-free automorphisms of prime order”, Proc. Nat. Acad. Sci. U.S.A., 45 (1959), 578–581 | DOI | MR | Zbl

[38] J. Thompson, “Automorphisms of solvable groups”, J. Algebra, 1 (1964), 259–267 | DOI | MR | Zbl

[39] A. Turull, “Fitting height of groups and of fixed points”, J. Algebra, 86 (1984), 555–566 | DOI | MR | Zbl