$\omega\sigma$-fibered Fitting classes

O. V. Kamozina

Čebyševskij sbornik, Tome 21 (2020) no. 4, pp. 107-116.

Résumé

The paper considers only finite groups. A class of groups $\mathfrak F$ is called a Fitting class if it is closed under normal subgroups and products of normal $\mathfrak F$-subgroups; formation, if it is closed with respect to factor groups and subdirect products; Fitting formation if $\mathfrak F$ is a formation and Fitting class at the same time. For a nonempty subset $\omega$ of the set of primes $\mathbb P$ and the partition $\sigma =\{\sigma_i\mid i\in I\}$, where $\mathbb P=\cup_{i\in I}\sigma _i$ and $\sigma_i\cap\sigma_j=\varnothing$ for all $i\not =j$, we introduce the $\omega\sigma R$-function $f$ and $\omega\sigma FR$-function $\varphi$. The domain of these functions is the set $\omega\sigma\cup\{\omega'\}$, where $\omega\sigma=\{ \omega\cap\sigma_i\mid\omega\cap\sigma_i\not =\varnothing\}$, $\omega'=\mathbb P\setminus\omega$. The range of function values is the set of Fitting classes and the set of nonempty Fitting formations, respectively. The functions $f$ and $\varphi$ are used to determine the $\omega\sigma$-fibered Fitting class $\mathfrak F=\omega\sigma R(f,\varphi)=(G: O^{\omega} (G)\in f(\omega' )$ and $G^{\varphi (\omega\cap\sigma_i )} \in f(\omega\cap\sigma_i )$ for all $\omega\cap\sigma_i \in\omega\sigma (G))$ with the $\omega\sigma$-satellite $f$ and the $\omega\sigma$-direction $\varphi$. The paper gives examples of $\omega\sigma$-fibered Fitting classes. Two types of $\omega\sigma$-fibered Fitting classes are distinguished: $\omega\sigma$-complete and $\omega\sigma$-local Fitting classes. Their directions are indicated by $\varphi_0$ and $\varphi_1$, respectively. It is shown that each nonempty nonidentity Fitting class is an $\omega\sigma$-complete Fitting class for some nonempty set $\omega\subseteq\mathbb P$ and any partition $\sigma$. A number of properties of $\omega\sigma$-fibered Fitting classes are obtained. In particular, a definition of an internal $\omega\sigma$-satellite is given and it is shown that each $\omega\sigma$-fibered Fitting class always has an internal $\omega\sigma$-satellite. For $\omega=\mathbb P$, the concept of a $\sigma$-fibered Fitting class is introduced. The connection between $\omega\sigma$-fibered and $\sigma$-fibered Fitting classes is shown.

Détail

Keywords: finite group, Fitting class, $\omega\sigma$-fibered, $\omega\sigma$-satellite, $\omega\sigma$-direction.

Bibliographie

[1] Gaschutz W., “Zur Theorie der endlichen auflosbaren Gruppen”, Math. Z., 80:4 (1963), 300–305 | MR | Zbl

[2] Hartley V., “On Fischer's dualization of formation theory”, Proc. London Math. Soc., 3:2 (1969), 193–207 | DOI | MR | Zbl

[3] Shemetkov L. A., Finite group formations, Nauka Publ., 1978, 272 pp. (in Russ.) | MR

[4] Vorob'ev N. T., “About local radical classes”, Voprosy algebry, 1986, no. 1, 22–34 (in Russ.)

[5] Skiba A. N., Algebra of formations, Belaruskaya navuka Publ., Minsk, 1997, 240 pp. (in Russ.) | MR

[6] Skiba A. N., Shemetkov L. A., “Multiply $\omega$-local formations and Fitting classes of finite groups”, Siberian Adv. Math., 10:2 (2000), 112–141 | MR | MR | Zbl | Zbl

[7] Vedernikov V. A., Sorokina M. M., “$\omega$-fibered formations and Fitting classes of finite groups”, Mathematical Notes, 71:1 (2002), 39–55 | DOI | MR | Zbl

[8] Shemetkov L. A., Skiba A. N., Formations of algebraic systems, Nauka Publ., M., 1989, 256 pp. (in Russ.) | MR

[9] Doerk K., Nawkes T., Finite soluble groups, Walter de Gruyter, Berlin-New York, 1992, 892 pp. | MR

[10] Vedernikov V .A., “On new types of $\omega$-fibered Fitting classes of finite groups”, Ukrainian Mathematical Journal, 54:7 (2002), 897–906 | DOI | MR | Zbl

[11] Korpacheva M. A., Sorokina M. M., “On critical $\omega$-fibered formations of finite groups”, Mathematical Notes, 79:1 (2006), 79–85 | DOI | DOI | MR | Zbl

[12] Safonov V. G., “On a question of the theory of totally local formations of finite groups”, Algebra and Logic, 42:6 (2003), 407–412 | DOI | MR | Zbl

[13] Vorob'ev N. N., Algebra of classes of finite groups, VGU imeni P. M. Masherova Publ., Vitebsk, 2012, 322 pp. (in Russ.)

[14] Guo Wenbin, Structure theory for canonical classes of finite groups, Springer-Verlag, Berlin–Heidelberg, 2015, 360 pp. | MR | Zbl

[15] Skiba A. N., “On one generalization of the local formations”, Problems of Physics, Mathematics and Technics, 2018, no. 1 (34), 79–82 | MR | Zbl