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On $$\mathfrak F_n$$-normal subgroups of finite groups. (English. Russian original) Zbl 1230.20015
Sib. Math. J. 52, No. 2, 197-206 (2011); translation from Sib. Mat. Zh. 52, No. 2, 250-264 (2011).
All groups considered are finite. For a class $$\mathfrak F$$ of groups, in the paper under review a subgroup $$H$$ of a group $$G$$ is defined to be $$\mathfrak F_n$$-normal in $$G$$ if there exists a normal subgroup $$T$$ of $$G$$ such that $$HT$$ is a normal subgroup of $$G$$ and $$(H\cap T)H_G/H_G$$ is contained in the $$\mathfrak F$$-hypercenter $$Z_\infty^{\mathfrak F}(G/H_G)$$ of $$G/H_G$$, where $$H_G$$ denotes the core of $$H$$ in $$G$$. We recall that the $$\mathfrak F$$-hypercenter is the largest normal subgroup of $$G$$ whose $$G$$-chief factors are $$\mathfrak F$$-central (a chief factor $$H/K$$ of $$G$$ is $$\mathfrak F$$-central if $$[H/K](G/C_G(H/K))\in\mathfrak F$$).
This concept is a generalization of $$c$$-normality (introduced by Y. Wang [in J. Pure Appl. Algebra 110, No. 3, 315-320 (1996; Zbl 0853.20015)]), $$\mathfrak F_n$$-supplementation (or $$\mathfrak F_c$$-normality) (considered by N. Yang and the first author [in Asian-Eur. J. Math. 1, No. 4, 619-629 (2008; Zbl 1176.20018)] and by A. Y. Alsheik Ahmad, J. J. Jaraden and A. N. Skiba [in Algebra Colloq. 14, No. 1, 25-36 ( 2007; Zbl 1126.20012)]) and $$\mathfrak F_h$$-normality (defined by X. Feng and the first author [in Front. Math. China 5, No. 4, 653-664 (2010; Zbl 1226.20011)]).
In this paper the authors use this new embedding property to obtain some criteria for supersolubility and $$p$$-nilpotency ($$p$$ a prime) of groups, taking further some previous developments.

##### MSC:
 20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, $$\pi$$-length, ranks 20D15 Finite nilpotent groups, $$p$$-groups 20D40 Products of subgroups of abstract finite groups
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