Abstract
In the paper, the notion of an \(\mathfrak F^{\omega}\)-abnormal (and \(\mathfrak F^{\omega}\)-normal) maximal subgroup of a finite group is introduced, where \(\mathfrak F\) is a nonempty class of groups and \(\omega\) is a nonempty set of primes. The relationship between the \(\mathfrak F^{\omega}\)- abnormal maximal and normal subgroups is studied. Conditions are established under which \(\mathfrak F^{\omega}\)-abnormal maximal subgroups in a finite group are normal.
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Acknowledgments
The authors of the paper are grateful to the referee for an important remark about the inexpediency of using the notion of a quasisubnormal subgroup due to the positive solution of the Wielandt–Kegel problem about the subnormality of any quasisubnormal subgroup in a group, which was obtained by Kleidman in [23]. In this connection, the authors replaced the condition “is quasisubnormal in \(G\)” for the maximal subgroups under consideration in the initial versions of Theorems 1 and 2 by the condition “is normal in \(G\).”
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Sorokina, M.M., Maksakov, S.P. On the Normality of \(\mathfrak F^{\omega}\)-Abnormal Maximal Subgroups of Finite Groups. Math Notes 108, 409–418 (2020). https://doi.org/10.1134/S0001434620090096
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DOI: https://doi.org/10.1134/S0001434620090096