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 Algebra Logika, 2013, Volume 52, Number 1, Pages 99–108 (Mi al575)

Rank and order of a finite group admitting a Frobenius group of automorphisms

E. I. Khukhro

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

Abstract: Suppose that a finite group $G$ admits a Frobenius group $FH$ of automorphisms of coprime order with kernel $F$ and complement $H$. For the case where $G$ is a finite $p$-group such that $G=[G,F]$, it is proved that the order of $G$ is bounded above in terms of the order of $H$ and the order of the fixed-point subgroup $C_G(H)$ of the complement, while the rank of $G$ is bounded above in terms of $|H|$ and the rank of $C_G(H)$. Earlier, such results were known under the stronger assumption that the kernel $F$ acts on $G$ fixed-point-freely. As a corollary, for the case where $G$ is an arbitrary finite group with a Frobenius group $FH$ of automorphisms of coprime order with kernel $F$ and complement $H$, estimates are obtained which are of the form $|G|\le|C_G(F)|\cdot f(|H|,|C_G(H)|)$ for the order, and of the form $\mathbf r(G)\le\mathbf r(C_G(F))+g(|H|,\mathbf r(C_G(H)))$ for the rank, where $f$ and $g$ are some functions of two variables.

Keywords: finite group, Frobenius group, automorphism, rank, order, $p$-group.

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English version:
Algebra and Logic, 2013, 52:1, 72–78

Bibliographic databases:

UDC: 512.542

Citation: E. I. Khukhro, “Rank and order of a finite group admitting a Frobenius group of automorphisms”, Algebra Logika, 52:1 (2013), 99–108; Algebra and Logic, 52:1 (2013), 72–78

Citation in format AMSBIB
\Bibitem{Khu13} \by E.~I.~Khukhro \paper Rank and order of a~finite group admitting a~Frobenius group of automorphisms \jour Algebra Logika \yr 2013 \vol 52 \issue 1 \pages 99--108 \mathnet{http://mi.mathnet.ru/al575} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3113801} \zmath{https://zbmath.org/?q=an:06189477} \transl \jour Algebra and Logic \yr 2013 \vol 52 \issue 1 \pages 72--78 \crossref{https://doi.org/10.1007/s10469-013-9221-1} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000319133000008} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84877085218} 

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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. İ. Güloğlu, G. Ercan, “Action of a Frobenius-like group”, J. Algebra, 402 (2014), 533–543
2. G. Ercan, İ. Güloğlu, E. I. Khukhro, “Rank and order of a finite group admitting a Frobenius-like group of automorphisms”, Algebra and Logic, 53:3 (2014), 258–265
3. G. Ercan, İ. Güloğlu, E. I. Khukhro, “Derived length of a Frobenius-like kernel”, J. Algebra, 412 (2014), 179–188
4. G. Ercan, İ. Güloğlu, E. I. Khukhro, “Frobenius-like groups as groups of automorphisms”, Turk. J. Math., 38:6 (2014), 965–976
5. E. I. Khukhro, N. Yu. Makarenko, “Finite $p$-groups with a Frobenius group of automorphisms whose kernel is a cyclic $p$-group”, Proc. Amer. Math. Soc., 143:5 (2015), PII S0002-9939(2015)12287-3, 1837–1848
6. Gülin Ercan, İsmail Ş. Güloğlu, “Finite groups admitting a dihedral group of automorphisms”, Algebra Discrete Math., 23:2 (2017), 223–229
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