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

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

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
\Bibitem{MakKhu06} \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 \mathnet{http://mi.mathnet.ru/al160} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2307695} \zmath{https://zbmath.org/?q=an:1156.20022} \elib{https://elibrary.ru/item.asp?id=9462657} \transl \jour Algebra and Logic \yr 2006 \vol 45 \issue 5 \pages 326--343 \crossref{https://doi.org/10.1007/s10469-006-0030-7} \elib{https://elibrary.ru/item.asp?id=13533192} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33750739790} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
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
3. Khukhro E.I., Makarenko N.Yu., “Automorphically-invariant ideals satisfying multilinear identities, and group-theoretic applications”, J. Algebra, 320:4 (2008), 1723–1740
4. Khukhro E.I., “Large normal and characteristic subgroups satisfying outer commutator identities and their applications”, Ischia: Group Theory 2008, 2009, 131–155
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
6. A. V. Akishin, “On groups with automorphisms generating recurrent sequences of the maximal period”, Discrete Math. Appl., 25:4 (2015), 187–192
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
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
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
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
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