V.N. Ryzhik, I.N. Safonova, A.N. Skiba. On the $\mathfrak{F}$-norm of a finite group ... P. 232-238

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.

Keywords: finite group, hereditary formation, $\mathfrak{F}$-residual of a group, $\mathfrak{F}$-norm of a group, generalized $\mathfrak{F}$-critical group

Received November 10, 2021

Revised December 15, 2021

Accepted December 27, 2021

Funding Agency: 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).

Valentina Nikolaevna Rizhik, Cand. Sci. (Phys.-Math.), Department of Automation, Physics and Mathematics, Bryansk State Agrarian University, Bryansk, 243365, Russia, e-mail: v2929222@yandex.ru

Inna Nikolaevna Safonova, Cand. Sci. (Phys.-Math.), Faculty of Applied Mathematics and Computer Science, Belarusian State University, Minsk, 220030, Belarus, e-mail: safonova@bsu.by

Alexander Nikolaevich Skiba, Department of Mathematics and Programming Technologies, Francisk Skorina Gomel State University, Gomel, 246019, Belarus, e-mail: alexander.skiba49@gmail.com

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Cite this article as: V.N. Ryzhik, I.N. Safonova, A.N. Skiba. On the $\mathfrak{F}$-norm of a finite group, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2022, vol. 28, no. 1, pp. 232–238; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2022, Vol. 317, Suppl. 1, pp. S136–S141.