On new classes of conjugate injectors of finite groups

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.