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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
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
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
Linking options:
https://www.mathnet.ru/eng/al17 https://www.mathnet.ru/eng/al/v42/i1/p51
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