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On finite groups with $Q$-central elements of prime order
O. L. Shemetkova
Plekhanov Russian State Academy of Economics, Moscow
Following L. A. Shemetkov, an element $x$ of a non-nilpotent finite group $X$ is called a $Q$-central element if there exists a central chief factor $H/L$ of $X$ such that $x\in H$ and $x\notin L$. An element $x$ is called a $Q_8$-element in a group if there exists a section $A/B$ such that $A/B$ contains $xB$ and is isomorphic to the quaternion group $Q_8$ of order $8$, and $o(x)$ coincides with the order of $xB$ in $A/B$. Let $G$ be a finite group such that every its element of prime order is $Q$-central. Then the following conditions hold: 1) a Sylow 2-subgroup $G_2$ of $G$ is normal and $G/G_2$ is nilpotent; 2) there is a $Q_8$-element in $G_2$ which is not $Q$-central in $G$.
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O. L. Shemetkova, “On finite groups with $Q$-central elements of prime order”, Tr. Inst. Mat., 16:1 (2008), 97–99
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\paper On finite groups with $Q$-central elements of prime order
\jour Tr. Inst. Mat.
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N. Yang, W. Guo, O. L. Shemetkova, “Finite groups with $S$-supplemented $p$-subgroups”, Siberian Math. J., 53:2 (2012), 371–376
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