Abstract
Let \(G\) be a finite group, and let \(\mathfrak{F}\) be a nonempty formation. Then the intersection of the normalizers of the \(\mathfrak{F}\)-residuals of all subgroups of \(G\) is called the \(\mathfrak{F}\)-norm of \(G\) and is denoted by \(N_{\mathfrak{F}}(G)\). A group \(G\) is called \(\mathfrak{F}\)-critical  if \(G\not\in\mathfrak{F}\), but \(U\in\mathfrak{F}\) for any proper subgroup \(U\) of \(G\). We say that a finite group \(G\) is generalized \(\mathfrak{F}\)-critical  if \(G\) contains a normal subgroup \(N\) such that \(N\leq\Phi(G)\) and the quotient group \(G/N\) is \(\mathfrak{F}\)-critical. In this publication, we prove the following result: If \(G\) does not belong to the nonempty hereditary formation \(\mathfrak{F},\) then the \(\mathfrak{F}\)-norm \(N_{\mathfrak{F}}(G)\) of \(G\) coincides with the intersection of the normalizers of the \(\mathfrak{F}\)-residuals of all generalized \(\mathfrak{F}\)-critical subgroups of \(G\). In particular\(,\) the norm \(N(G)\) of \(G\) coincides with the intersection of the normalizers of all cyclic subgroups of \(G\) of prime power order.
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Funding
The second author was supported by the Ministry of Education of the Republic of Belarus (project no. 20211328), and the third author was supported by the Belarusian Republican Foundation for Fundamental Research (grant no. F20R-291).
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Translated from Trudy Instituta Matematiki i Mekhaniki UrO RAN, Vol. 28, No. 1, pp. 232 - 238, 2022 https://doi.org/10.21538/0134-4889-2022-28-1-232-238.
Translated by E. Vasil’eva
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Ryzhik, V.N., Safonova, I.N. & Skiba, A.N. On the \(\mathfrak{F}\)-Norm of a Finite Group. Proc. Steklov Inst. Math. 317 (Suppl 1), S136–S141 (2022). https://doi.org/10.1134/S0081543822030129
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DOI: https://doi.org/10.1134/S0081543822030129