On new classes of conjugate injectors of finite groups

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 ℜ∗-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 𐔉 = 𐕳𐔅, where 𐕳 is a non-empty Fitting class and 𐔅 is a Blessenohl-Laue class, and thus we distinguish a new class of conjugate 𐔉-injectors in the classes 𐔈 of all finite groups and 𐔖π of all finite π-solvable groups respectively. Moreover, we prove that the 𐔉-injectors of the group G are exactly all 𐔉-maximal subgroups of G, which contain its 𐔉-radical G𐔉. Special cases of such injectors are the injectors for many known Fitting classes. In particular, such injectors in the class 𐔖 of all finite solvable groups were described by B. Hartley, B. Fischer, W. Frantz, and P. Lockett.