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Algebra Logika, 2006, Volume 45, Number 5, Pages 575–602 (Mi al160)  

This article is cited in 10 scientific papers (total in 10 papers)

Finite groups with an almost regular automorphism of order four

N. Yu. Makarenko, E. I. Khukhro

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences

Abstract: P. Shumyatsky's question 11.126 in the “Kourovka Notebook” is answered in the affirmative: it is proved that there exist a constant $c$ and a function of a positive integer argument $f(m)$ such that if a finite group $G$ admits an automorphism $\varphi$ of order 4 having exactly $m$ fixed points, then $G$ has a normal series $G\geqslant H\geqslant N$ such that $|G/H|\leqslant f(m)$, the quotient group $H/N$ is nilpotent of class $\leqslant 2$, and the subgroup $N$ is nilpotent of class $\leqslant c$ (Thm. 1). As a corollary we show that if a locally finite group $G$ contains an element of order 4 with finite centralizer of order $m$, then $G$ has the same kind of a series as in Theorem 1. Theorem 1 generalizes Kovác's theorem on locally finite groups with a regular automorphism of order 4, whereby such groups are center-by-metabelian. Earlier, the first author proved that a finite 2-group with an almost regular automorphism of order 4 is almost center-by-metabelian. The proof of Theorem 1 is based on the author's previous works dealing in Lie rings with an almost regular automorphism of order 4. Reduction to nilpotent groups is carried out by using Hall-Higman type theorems. The proof also uses Theorem 2, which is of independent interest, stating that if a finite group $S$ contains a nilpotent subgroup $T$ of class $c$ and index $|S:T|=n$, then $S$ contains also a characteristic nilpotent subgroup of class $\leqslant c$ whose index is bounded in terms of $n$ and $c$. Previously, such an assertion has been known for Abelian subgroups, that is, for $c=1$.

Keywords: finite group, almost regular automorphism, Lie ring, nilpotency class, centralizer, Hall-Higman type theorems, characteristic subgroup

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English version:
Algebra and Logic, 2006, 45:5, 326–343

Bibliographic databases:

UDC: 512.54
Received: 31.05.2006

Citation: N. Yu. Makarenko, E. I. Khukhro, “Finite groups with an almost regular automorphism of order four”, Algebra Logika, 45:5 (2006), 575–602; Algebra and Logic, 45:5 (2006), 326–343

Citation in format AMSBIB
\by N.~Yu.~Makarenko, E.~I.~Khukhro
\paper Finite groups with an almost regular automorphism of order four
\jour Algebra Logika
\yr 2006
\vol 45
\issue 5
\pages 575--602
\jour Algebra and Logic
\yr 2006
\vol 45
\issue 5
\pages 326--343

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    This publication is cited in the following articles:
    1. Khukhro E.I., Makarenko N.Yu., “Large characteristic subgroups satisfying multilinear commutator identities”, J. Lond. Math. Soc. (2), 75:3 (2007), 635–646  crossref  mathscinet  zmath  isi  scopus
    2. Shumyatsky P., “On the centralizer of an element of order four in a locally finite group”, Glasg. Math. J., 49:2 (2007), 411–415  crossref  mathscinet  zmath  isi  elib  scopus
    3. Khukhro E.I., Makarenko N.Yu., “Automorphically-invariant ideals satisfying multilinear identities, and group-theoretic applications”, J. Algebra, 320:4 (2008), 1723–1740  crossref  mathscinet  zmath  isi  elib  scopus
    4. Khukhro E.I., “Large normal and characteristic subgroups satisfying outer commutator identities and their applications”, Ischia: Group Theory 2008, 2009, 131–155  crossref  mathscinet  zmath  isi
    5. Evgeny I. Khukhro, “Problems of bounding the $p$-length and Fitting height of finite soluble groups”, Zhurn. SFU. Ser. Matem. i fiz., 6:4 (2013), 462–478  mathnet
    6. A. V. Akishin, “On groups with automorphisms generating recurrent sequences of the maximal period”, Discrete Math. Appl., 25:4 (2015), 187–192  mathnet  crossref  crossref  mathscinet  isi  elib
    7. A. V. Akishin, “On groups of even orders with automorphisms generating recurrent sequences of the maximal period”, Discrete Math. Appl., 25:5 (2015), 253–259  mathnet  crossref  crossref  mathscinet  isi  elib
    8. E. I. Khukhro, “On finite soluble groups with almost fixed-point-free automorphisms of noncoprime order”, Siberian Math. J., 56:3 (2015), 541–548  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    9. Khukhro E.I., Makarenko N.Yu., Shumyatsky P., “Locally Finite Groups Containing a -Element With Chernikov Centralizer”, Mon.heft. Math., 179:1 (2016), 91–97  crossref  mathscinet  zmath  isi  scopus
    10. Khukhro E.I., Makarenko N.Yu., Shumyatsky P., “Finite Groups and Lie Rings With An Automorphism of Order 2N”, Proc. Edinb. Math. Soc., 60:2 (2017), 391–412  crossref  mathscinet  zmath  isi  scopus
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