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On new classes of conjugate injectors of finite groups. (English. Russian original) Zbl 1065.20031
Discrete Math. Appl. 14, No. 2, 191-199 (2004); translation from Diskretn. Mat. 16, No. 1, 105-113 (2004).
The present paper generalises a result by D. Blessenohl and H. Laue [J. Algebra 56, 516–532 (1979; Zbl 0416.20015)] on quasinilpotent injectors in finite groups. In that paper, two classes of groups, \(\mathfrak B\) and \(\mathfrak C\), were introduced, which are now called \(BL\)-classes. If \(\mathfrak H\) is a non-empty Fitting class, \({\mathfrak F}=\mathfrak{HB}\) and \(G\) is a finite group such that \(C_{G/G_{\mathfrak H}}((G/G_{\mathfrak H})_{\mathfrak B})\leq(G/G_{\mathfrak H})_{\mathfrak B}\), then (i) \(G\) has \(\mathfrak F\)-injectors and any two of them are conjugate in \(G\), (ii) the \(\mathfrak F\)-injectors of \(G\) are exactly those \(\mathfrak F\)-maximal subgroups of \(G\) which contain \(G_{\mathfrak F}\) (Theorem 1). If \({\mathfrak F}=\mathfrak{HC}\) and \(G\) is a finite \(\pi\)-soluble group such that \(C_{G/G_{\mathfrak H}}((G/G_{\mathfrak H})_{\mathfrak C})\leq (G/G_{\mathfrak H})_{\mathfrak C}\), then (i) \(G\) has \(\mathfrak F\)-injectors and any two of them are conjugate in \(G\), (ii) the \(\mathfrak F\)-injectors of \(G\) are exactly those \(\mathfrak F\)-maximal subgroups of \(G\) which contain \(G_{\mathfrak F}\) (Theorem 2). The two theorems are followed by twelve corollaries.
MSC:
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D25 Special subgroups (Frattini, Fitting, etc.)
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