Geodesic
Permuteral subgroups and their applications in finite groups
Problemy fiziki, matematiki i tehniki, no. 2 (2013), pp. 35-38.

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Let $H$ be a subgroup of a group $G$. The permutizer of $H$ in $G$ is the subgroup $P_G(H)=\langle x\in G | \langle x\rangle H=H\langle x\rangle\rangle$. The subgroup $H$ of a group $G$ is called permuteral in $G$, if $P_G(H)=G$; strongly permuteral in $G$, if $P_U(H)=U$ whenever $H\leqslant U\leqslant G$. The properties of finite groups with given systems of permuteral and strongly permuteral subgroups are obtained. New criteria of w-supersolubility and supersolubility of groups are received.
Keywords: finite group, permutizer of a subgroup, permuteral subgroup, supersoluble group, w-supersoluble group, $\mathbf{P}$-subnormal subgroup. 
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A. F. Vasil'ev; V. A. Vasil'ev; T. I. Vasil'eva. Permuteral subgroups and their applications in finite groups. Problemy fiziki, matematiki i tehniki, no. 2 (2013), pp. 35-38. http://geodesic.mathdoc.fr/item/PFMT_2013_2_a5/

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