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 Diskr. Mat., 2004, Volume 16, Issue 1, Pages 105–113 (Mi dm145)

On new classes of conjugate injectors of finite groups

E. N. Zalesskaya

Abstract: In the study of the problem of existence and conjugacy in an arbitrary finite group it is known the Blessenohl–Laue result that in any finite group $G$ there exists a unique class of conjugate quasinilpotent injectors which are exactly the $\mathfrak N^*$-maximal subgroups of $G$ containing the generalised Fitting subgroup $F^*(G)$. In this paper, with the use of constructions of the Blessenohl–Laue and Gaschütz classes, we extend the Blessenohl–Laue result to the case of the Fitting class $\mathfrak F=\mathfrak H\mathfrak B$, where $\mathfrak H$ is a non-empty Fitting class and $\mathfrak B$ is a Blessenohl–Laue class, and thus we distinguish a new class of conjugate $\mathfrak F$-injectors in the classes $\mathfrak E$ of all finite groups and $\mathfrak S^{\pi}$ of all finite $\pi$-solvable groups respectively. Moreover, we prove that the $\mathfrak F$-injectors of the group $G$ are exactly all $\mathfrak F$-maximal subgroups of $G$, which contain its $\mathfrak F$-radical $G_{\mathfrak F}$. Special cases of such injectors are the injectors for many known Fitting classes. In particular, such injectors in the class $\mathfrak S$ of all finite solvable groups were described by B. Hartley, B. Fischer, W. Frantz, and P. Lockett.

DOI: https://doi.org/10.4213/dm145

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English version:
Discrete Mathematics and Applications, 2004, 14:2, 191–199

Bibliographic databases:

UDC: 512.542

Citation: E. N. Zalesskaya, “On new classes of conjugate injectors of finite groups”, Diskr. Mat., 16:1 (2004), 105–113; Discrete Math. Appl., 14:2 (2004), 191–199

Citation in format AMSBIB
\Bibitem{Zal04} \by E.~N.~Zalesskaya \paper On new classes of conjugate injectors of finite groups \jour Diskr. Mat. \yr 2004 \vol 16 \issue 1 \pages 105--113 \mathnet{http://mi.mathnet.ru/dm145} \crossref{https://doi.org/10.4213/dm145} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2069992} \zmath{https://zbmath.org/?q=an:1065.20031} \transl \jour Discrete Math. Appl. \yr 2004 \vol 14 \issue 2 \pages 191--199 \crossref{https://doi.org/10.1515/156939204872338} 

• http://mi.mathnet.ru/eng/dm145
• https://doi.org/10.4213/dm145
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1. Goiko V.I., “In'ektory konechnykh grupp”, Vesn³k V³tsebskaga dzyarzhainaga un³vers³teta, 1:67 (2012), 5–11
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