Geodesic
On $MP$-closed saturated formations of finite groups
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 6 (2017), pp. 9-17

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A class of groups $\mathfrak{F}$ is called $MP$-closed, if it contains every group $G=AB$ such that $\mathfrak{F}$-subgroup $A$ permutes with every subgroup of $B$ and $\mathfrak{F}$-subgroup $B$ permutes with every subgroup of $A$. We prove that the formation $\mathfrak{F}$ containing the class of all supersoluble groups is $MP$-closed if and only if the formation $F(p)$ is $MP$-closed for all prime $p$, where $F$ is maximal integrated local screen of $\mathfrak{F}$. In particular, we prove that the formation of all groups with supersoluble Schmidt subgroups is $MP$-closed.
Keywords: finite group, product of mutually permutable subgroups, saturated formation, $MP$-closed formation, local screen. 
@article{IVM_2017_6_a1,
     author = {A. F. Vasil'ev and T. I. Vasil'eva and D. N. Simonenko},
     title = {On $MP$-closed saturated formations of finite groups},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {9--17},
     publisher = {mathdoc},
     number = {6},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2017_6_a1/}
}
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A. F. Vasil'ev; T. I. Vasil'eva; D. N. Simonenko. On $MP$-closed saturated formations of finite groups. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 6 (2017), pp. 9-17. http://geodesic.mathdoc.fr/item/IVM_2017_6_a1/