
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 centerbymetabelian. Earlier, the first author proved that a finite 2group with an almost regular automorphism of order 4 is almost centerbymetabelian. 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 HallHigman 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, HallHigman type theorems, characteristic subgroup
Full text:
PDF file (295 kB)
References:
PDF file
HTML file
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
\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 575602
\mathnet{http://mi.mathnet.ru/al160}
\mathscinet{http://www.ams.org/mathscinetgetitem?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 326343
\crossref{https://doi.org/10.1007/s1046900600307}
\elib{https://elibrary.ru/item.asp?id=13533192}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2s2.033750739790}
Linking options:
http://mi.mathnet.ru/eng/al160 http://mi.mathnet.ru/eng/al/v45/i5/p575
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:

Khukhro E.I., Makarenko N.Yu., “Large characteristic subgroups satisfying multilinear commutator identities”, J. Lond. Math. Soc. (2), 75:3 (2007), 635–646

Shumyatsky P., “On the centralizer of an element of order four in a locally finite group”, Glasg. Math. J., 49:2 (2007), 411–415

Khukhro E.I., Makarenko N.Yu., “Automorphicallyinvariant ideals satisfying multilinear identities, and grouptheoretic applications”, J. Algebra, 320:4 (2008), 1723–1740

Khukhro E.I., “Large normal and characteristic subgroups satisfying outer commutator identities and their applications”, Ischia: Group Theory 2008, 2009, 131–155

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

A. V. Akishin, “On groups with automorphisms generating recurrent sequences of the maximal period”, Discrete Math. Appl., 25:4 (2015), 187–192

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

E. I. Khukhro, “On finite soluble groups with almost fixedpointfree automorphisms of noncoprime order”, Siberian Math. J., 56:3 (2015), 541–548

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

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

Number of views: 
This page:  343  Full text:  80  References:  42  First page:  3 
