Geodesic
Groups Containing a Self-Centralizing Subgroup of Order~3
Algebra i logika, Tome 42 (2003) no. 1, pp. 51-64.

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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.
Mots-clés : group, Frobenius group
Keywords: centralizer, conjugate subgroup, normal subgroup, nilpotent subgroup, field. 
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V. D. Mazurov. Groups Containing a Self-Centralizing Subgroup of Order~3. Algebra i logika, Tome 42 (2003) no. 1, pp. 51-64. http://geodesic.mathdoc.fr/item/AL_2003_42_1_a3/

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