Satellites and products of $\omega\sigma$-fibered Fitting classes

O. V. Kamozina

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 27 (2021) no. 1, pp. 88-97

Résumé

A Fitting class $\frak F=\omega\sigma R(f,\varphi)=(G: O^\omega (G)\in f(\omega')$ and $G^{\varphi(\omega\cap\sigma_i)}\in f(\omega\cap\sigma_i)$ for all $\omega\cap\sigma_i\in\omega\sigma (G))$ is called an $\omega\sigma$-fibered Fitting class with $\omega\sigma$-satellite $f$ and $\omega\sigma$-direction $\varphi$. By $\varphi_0$ and $\varphi_1$ we denote the directions of an $\omega\sigma$-complete and an $\omega\sigma$-local Fitting class, respectively. Theorem 1 describes a minimal $\omega\sigma$-satellite of an $\omega\sigma$-fibered Fitting class with $\omega\sigma$-direction $\varphi$, where $\varphi_0\le\varphi$. Theorem 2 states that the Fitting product of two $\omega\sigma$-fibered Fitting classes is an $\omega\sigma$-fibered Fitting class for $\omega\sigma$-directions $\varphi$ such that $\varphi_0\le\varphi\le\varphi_1$. Results for $\omega\sigma$-complete and $\omega\sigma$-local Fitting classes are obtained as corollaries of the theorems. Theorem 3 describes a maximal internal $\omega\sigma$-satellite of an $\omega\sigma$-complete Fitting class. An $\omega\sigma\mathcal L$-satellite is defined as an $\omega\sigma$-satellite $f$ such that $f(\omega\cap\sigma_i)$ is the Lockett class for all $\omega\cap\sigma_i \in\omega\sigma$. Theorem 4 describes the maximal internal $\omega\sigma\mathcal L$-satellite of an $\omega\sigma$-local Fitting class. Questions of the study of lattices and further study of products and critical $\omega\sigma$-fibered Fitting classes are posed in the conclusion.

Détail

Keywords: finite group, Fitting class, $\omega\sigma$-fibered, $\omega\sigma$-complete, $\omega\sigma$-local, minimal $\omega\sigma$-satellite, Fitting product.
Mots-clés : maximal internal $\omega\sigma$-satellite

Bibliographie

[1] Vorobev N.N., Skiba A.N., “O distributivnosti reshetki razreshimykh totalno lokalnykh klassov Fittinga”, Mat. zametki, 67:5 (2000), 662–673 | MR | Zbl

[2] Safonov V.G., Safonova I.N., “O minimalnykh totalno lokalnykh ne $\pi$-nilpotentnykh klassakh Fittinga”, Izv. Gomel. gos. un-ta im. F. Skoriny, 2016, no. 6 (99), 91–95

[3] Egorova V.E., “Kriticheskie neodnoporozhdennye totalno kanonicheskie klassy Fittinga konechnykh grupp”, Mat. zametki, 83:4 (2008), 520–527 | Zbl

[4] Kamozina O.V., “$\omega$$\sigma$-veernye klassy Fittinga”, Chebysh. sb., 21:4 (2020), 107–116 | DOI | MR

[5] Skiba A.N., “On one generalization of the local formations”, Problemy fiziki, matematiki i tekhniki, 2018, no. 1(34), 79–82 | MR | Zbl

[6] Skiba A.N., Shemetkov L.A., “Kratno $\omega$-lokalnye formatsii i klassy Fittinga konechnykh grupp”, Mat. tr., 2:2 (1999), 114–147 | MR | Zbl

[7] Doerk K., Nawkes T., Finite soluble groups, Walter de Gruyter, Berlin; N Y, 1992, 892 pp. | MR

[8] Vedernikov V.A., “O novykh tipakh $\omega$-veernykh klassov Fittinga konechnykh grupp”, Ukrain. mat. zhurn., 54:7 (2002), 897–906 | DOI | MR | Zbl