Geodesic
Finite partially soluble groups with transitive $\pi$-quasinormality relation for subgroups
Trudy Instituta matematiki, Tome 31 (2023) no. 2, pp. 28-33

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Throughout the article, all groups are finite. We say that a subgroup $A$ of $G$ is $\pi$-quasinormal in $G$, if $A$ is $1 \pi$-subnormal and modular in $G$. We prove that if the group $G$ is $\pi _{0}$-solvable, where $\pi _{0}=\pi (D) $ and $D$ is the $\pi $-special residual of $G$, and $\pi$-quasi-normality is a transitive relation in $G$, then $D$ is an abelian Hall subgroup of odd order in $G$. 
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     author = {I. M. Dergacheva and E. A. Zadorozhnyuk and I. P. Shabalina},
     title = {Finite partially soluble groups with transitive $\pi$-quasinormality relation for subgroups},
     journal = {Trudy Instituta matematiki},
     pages = {28--33},
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     volume = {31},
     number = {2},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMB_2023_31_2_a3/}
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I. M. Dergacheva; E. A. Zadorozhnyuk; I. P. Shabalina. Finite partially soluble groups with transitive $\pi$-quasinormality relation for subgroups. Trudy Instituta matematiki, Tome 31 (2023) no. 2, pp. 28-33. http://geodesic.mathdoc.fr/item/TIMB_2023_31_2_a3/