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.
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
\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}
Linking options:
https://www.mathnet.ru/eng/al17
https://www.mathnet.ru/eng/al/v42/i1/p51
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Giudici M., Kuzma B., “Realizability Problem For Commuting Graphs”, J. Aust. Math. Soc., 101:3 (2016), 335–355
Jabara E., Lytkina D.V., Mazurov V.D., “Some Groups of Exponent 72”, J. Group Theory, 17:6 (2014), 947–955
Zhang J., Shen Zh., Shi J., “Finite Groups With Few Vanishing Elements”, Glas. Mat., 49:1 (2014), 83–103
Zhang J., Shi J., Shen Zh., “Finite groups whose irreducible characters vanish on at most three conjugacy classes”, Journal of Group Theory, 13:6 (2010), 799–819
Zhang J., Shen Zh., Liu D., “On Zeros of Characters of Finite Groups”, Czechoslovak Mathematical Journal, 60:3 (2010), 801–816
Astill S., Parker Ch., “A 3-Local Characterization of M-12 and SL3(3)”, Archiv der Mathematik, 92:2 (2009), 99–110
Zhang J., Shi W., “Two dual questions on zeros of characters of finite groups”, Journal of Group Theory, 11:5 (2008), 697–708
Mazurov VD, “Characterizations of groups by arithmetic properties”, Algebra Colloquium, 11:1 (2004), 129–140
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