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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.
UDC:
512.542 Received: 03.04.2003
DOI:
10.4213/dm145
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