Geodesic
On permutable strongly $2$-maximal and strongly $3$-maximal subgroups
Vestnik rossijskih universitetov. Matematika, Tome 26 (2021) no. 134, pp. 121-129

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We describe the structure of finite solvable non-nilpotent groups in which every two strongly $n$-maximal subgroups are permutable ($n = 2, 3$). In particular, it is shown for a solvable non-nilpotent group $G$ that any two strongly $2$-maximal subgroups are permutable if and only if $G$ is a Schmidt group with Abelian Sylow subgroups. We also prove the equivalence of the structure of non-nilpotent solvable groups with permutable $3$-maximal subgroups and with permutable strongly $3$-maximal subgroups. The last result allows us to classify all finite solvable groups with permutable strongly $3$-maximal subgroups, and we describe $14$ classes of groups with this property. The obtained results also prove the nilpotency of a finite solvable group with permutable strongly $n$-maximal subgroups if the number of prime divisors of the order of this group strictly exceeds $n$ ($n=2, 3$).
Mots-clés : solvable group
Keywords: $n$-maximal subgroup, strongly $n$-maximal subgroup, normal subgroup, nilpotent group, Schmidt group. 
@article{VTAMU_2021_26_134_a1,
     author = {Yu. V. Gorbatova},
     title = {On permutable strongly $2$-maximal and strongly $3$-maximal subgroups},
     journal = {Vestnik rossijskih universitetov. Matematika},
     pages = {121--129},
     publisher = {mathdoc},
     volume = {26},
     number = {134},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VTAMU_2021_26_134_a1/}
}
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Yu. V. Gorbatova. On permutable strongly $2$-maximal and strongly $3$-maximal subgroups. Vestnik rossijskih universitetov. Matematika, Tome 26 (2021) no. 134, pp. 121-129. http://geodesic.mathdoc.fr/item/VTAMU_2021_26_134_a1/