Geodesic
Finite groups with semi-subnormal Schmidt subgroups
Algebra and discrete mathematics, Tome 29 (2020) no. 1, pp. 66-73

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A Schmidt group is a non-nilpotent group in which every proper subgroup is nilpotent. A subgroup $A$ of a group $G$ is semi-normal in $G$ if there exists a subgroup $B$ of $G$ such that $G=AB$ and $AB_1$ is a proper subgroup of $G$ for every proper subgroup $B_1$ of $B$. If $A$ is either subnormal in $G$ or is semi-normal in $G$, then $A$ is called a semi-subnormal subgroup of $G$. In this paper, we establish that a group $G$ with semi-subnormal Schmidt $\{2,3\}$-subgroups is $3$-soluble. Moreover, if all 5-closed Schmidt $\{2,5\}$-subgroups are semi-subnormal in $G$, then $G$ is soluble. We prove that a group with semi-subnormal Schmidt subgroups is metanilpotent.
Keywords: finite soluble group, Schmidt subgroup, semi-normal subgroup, subnormal subgroup. 
@article{ADM_2020_29_1_a5,
     author = {V. N. Knyagina and V. S. Monakhov},
     title = {Finite groups with semi-subnormal {Schmidt} subgroups},
     journal = {Algebra and discrete mathematics},
     pages = {66--73},
     publisher = {mathdoc},
     volume = {29},
     number = {1},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2020_29_1_a5/}
}
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V. N. Knyagina; V. S. Monakhov. Finite groups with semi-subnormal Schmidt subgroups. Algebra and discrete mathematics, Tome 29 (2020) no. 1, pp. 66-73. http://geodesic.mathdoc.fr/item/ADM_2020_29_1_a5/