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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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References:
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