On new classes of conjugate injectors of finite groups

E. N. Zalesskaya

Diskretnaya Matematika, Tome 16 (2004) no. 1, pp. 105-113

Résumé

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.