Geodesic
On finite groups with $Q$-central elements of prime order
Trudy Instituta matematiki, Tome 16 (2008) no. 1, pp. 97-99

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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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     title = {On finite groups with $Q$-central elements of prime order},
     journal = {Trudy Instituta matematiki},
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     number = {1},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMB_2008_16_1_a16/}
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O. L. Shemetkova. On finite groups with $Q$-central elements of prime order. Trudy Instituta matematiki, Tome 16 (2008) no. 1, pp. 97-99. http://geodesic.mathdoc.fr/item/TIMB_2008_16_1_a16/