Geodesic
On new classes of conjugate injectors of finite groups
Diskretnaya Matematika, Tome 16 (2004) no. 1, pp. 105-113.

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{DM_2004_16_1_a7,
     author = {E. N. Zalesskaya},
     title = {On new classes of conjugate injectors of finite groups},
     journal = {Diskretnaya Matematika},
     pages = {105--113},
     publisher = {mathdoc},
     volume = {16},
     number = {1},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2004_16_1_a7/}
}
TY  - JOUR
AU  - E. N. Zalesskaya
TI  - On new classes of conjugate injectors of finite groups
JO  - Diskretnaya Matematika
PY  - 2004
SP  - 105
EP  - 113
VL  - 16
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DM_2004_16_1_a7/
LA  - ru
ID  - DM_2004_16_1_a7
ER  - 
%0 Journal Article
%A E. N. Zalesskaya
%T On new classes of conjugate injectors of finite groups
%J Diskretnaya Matematika
%D 2004
%P 105-113
%V 16
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DM_2004_16_1_a7/
%G ru
%F DM_2004_16_1_a7
E. N. Zalesskaya. On new classes of conjugate injectors of finite groups. Diskretnaya Matematika, Tome 16 (2004) no. 1, pp. 105-113. http://geodesic.mathdoc.fr/item/DM_2004_16_1_a7/

[1] Fischer B., Gaschutz W., Hartley B., “Injektoren endlicher auflosbarer Gruppen”, Math. Z., 102:5 (1967), 337–339 | DOI | MR | Zbl

[2] Doerk K., Hawkes T., Finite soluble groups, Walter de Gruyter, Berlin, 1992 | MR

[3] Shemetkov L. A., “O podgruppakh $\pi$-razreshimykh grupp”, Konechnye gruppy, Nauka i tekhnika, Minsk, 1975, 207–212

[4] Shemetkov L. A., “Nekotorye svoistva in'ektorov konechnykh grupp”, Izv. Gomelskogo gos. un-ta im. F. Skoriny, Voprosy algebry, 15:1 (1999), 5–13

[5] Sementovskii V. G., “$\triangle$-nilpotentnye in'ektory konechnykh grupp”, Voprosy algebry, 1 (1985), 72–86 | MR | Zbl

[6] Blessenohl D., Laue H., “Fittingklassen endlicher Gruppen in denen gewisse Haupfaktoren einfach sind”, J. Algebra, 56 (1979), 516–532 | DOI | MR | Zbl

[7] Huppert B., Blackburn N., Finite groups. III, Springer, Berlin, 1982 | MR

[8] Hartley B., “On Fisher's dualization of formation theory”, Proc. London Math. Soc., 3:2 (1969), 193–207 | DOI | MR

[9] Fischer B., Klassen konjugierter Untergruppen in endlichen auflosbaren Gruppen, Habilitationschrift, Univ. Frankfurt, 1966

[10] Frantz W., Spezielle Fittingklassen und ihre Injektoren, Diplomarbeit, Kiel, 1970

[11] Lokett P., On the theory of Fitting classes of finite soluble groups, Ph. D. Thesis, Univ. Warwick, 1971

[12] Shemetkov L. A., Formatsii konechnykh grupp, Nauka, Moskva, 1978 | MR | Zbl

[13] Iranzo M. J., Perez Monasor F., “$\mathfrak F$-constraint with respect to a Fitting class”, Arch. Math., 46 (1986), 205–210 | DOI | MR | Zbl

[14] Mann A., “Injectors and normal subgroups of finite groups”, Israel J. Math., 9 (1971), 554–558 | DOI | MR | Zbl

[15] Laue P., “Über nichtauflosbare normale Fittingklassen”, J. Algebra, 45 (1977), 274–283 | DOI | MR | Zbl