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Algebra i logika, 2003, Volume 42, Number 1, Pages 51–64 (Mi al17)  

This article is cited in 13 scientific papers (total in 13 papers)

Groups Containing a Self-Centralizing Subgroup of Order 3

V. D. Mazurov

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences
References:
Abstract: In 1962 Feit and Thompson obtained a description of finite groups containing a subgroup $X$ of order 3 which coincides with its centralizer. This result is carried over arbitrary groups with the condition that $X$ with every one of its conjugates generate a finite subgroup. We prove the following theorem.
Theorem. Suppose that a group $G$ contains a subgroup $X$ of order $3$ such that $C_G(X)=\langle X\rangle$. If, for every $g\in G$, the subgroup $\langle X,X^g\rangle$ is finite, then one of the following statements holds:
$(1)$ $G=NN_G(X)$ for a periodic nilpotent subgroup $N$ of class $2$, and $NX$ is a Frobenius group with core $N$ and complement $X$.
$(2)$ $G=NA$, where $A$ is isomorphic to $A_5\simeq SL_2(4)$ and $N$ is a normal elementary Abelian $2$-subgroup; here, $N$ is a direct product of order $16$ subgroups normal in $G$ and isomorphic to the natural $SL_2(4)$-module of dimension $2$ over a field of order $4$.
$(3)$ $G$ is isomorphic to $L_2(7)$.
In particular, $G$ is locally finite.
Keywords: group, centralizer, Frobenius group, conjugate subgroup, normal subgroup, nilpotent subgroup, field.
Received: 06.11.2002
English version:
Algebra and Logic, 2003, Volume 42, Issue 1, Pages 29–36
DOI: https://doi.org/10.1023/A:1022676707499
Bibliographic databases:
UDC: 512.542
Language: Russian
Citation: V. D. Mazurov, “Groups Containing a Self-Centralizing Subgroup of Order 3”, Algebra Logika, 42:1 (2003), 51–64; Algebra and Logic, 42:1 (2003), 29–36
Citation in format AMSBIB
\Bibitem{Maz03}
\by V.~D.~Mazurov
\paper Groups Containing a Self-Centralizing Subgroup of Order~3
\jour Algebra Logika
\yr 2003
\vol 42
\issue 1
\pages 51--64
\mathnet{http://mi.mathnet.ru/al17}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1988023}
\zmath{https://zbmath.org/?q=an:1035.20025}
\transl
\jour Algebra and Logic
\yr 2003
\vol 42
\issue 1
\pages 29--36
\crossref{https://doi.org/10.1023/A:1022676707499}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-1842462940}
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  • This publication is cited in the following 13 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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