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\( \Omega\zeta \)-foliated Fitting classes. (Russian. English summary) Zbl 1454.20026

Summary: All groups under consideration are assumed to be finite. For a nonempty subclass of \(\Omega\) of the class of all simple groups \(\mathfrak{I}\) and the partition \(\zeta =\{\zeta_i\mid i\in I\}\), where \(\zeta_i\) is a nonempty subclass of the class \(\mathfrak{I}\), \(\mathfrak{I} =\cup_{i\in I}\zeta_i\) and \(\zeta_i \cap \zeta_j = \emptyset\) for all \(i\not = j\), \(\Omega\zeta R\)-function \(f\) and \(\Omega\zeta FR\)-function \(\varphi\) are introduced. The domain of these functions is the set \(\Omega\zeta\cup \{\Omega'\}\), where \(\Omega\zeta =\{\Omega\cap\zeta_i\mid\Omega\cap\zeta_i\not =\emptyset\}\), \(\Omega'=\mathfrak{I}\setminus\Omega \). The scope of these 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 \(\Omega\zeta \)-foliated Fitting class \(\mathfrak{F}=\Omega\zeta R(f,\varphi )=(G: O^\Omega (G)\in f(\Omega')\) and \(G^{\varphi (\Omega\cap\zeta_i )}\in f(\Omega\cap\zeta_i )\) for all \(\Omega\cap\zeta_i \in\Omega\zeta (G))\) with \(\Omega\zeta \)-satellite \(f\) and \(\Omega\zeta \)-direction \(\varphi \). The paper gives examples of \(\Omega\zeta \)-foliated Fitting classes. Two types of \(\Omega\zeta \)-foliated Fitting classes are defined: \( \Omega\zeta \)-free and \(\Omega\zeta \)-canonical Fitting classes. Their directions are indicated by \(\varphi_0\) and \(\varphi_1\) respectively. It is shown that each non-empty non-identity Fitting class is a \(\Omega\zeta \)-free Fitting class for some non-empty class \(\Omega\subseteq\mathfrak{I}\) and any partition \(\zeta \). A series of properties of \(\Omega\zeta \)-foliated Fitting classes is obtained. In particular, the definition of internal \( \Omega\zeta \)-satellite is given and it is shown that every \(\Omega\zeta \)-foliated Fitting class has an internal \(\Omega\zeta \)-satellite. For \(\Omega=\mathfrak{I}\), the concept of a \(\zeta \)-foliated Fitting class is introduced. The connection conditions between \(\Omega\zeta \)-foliated and \(\zeta \)-foliated Fitting classes are shown.

MSC:

20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
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References:

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