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On $$\mathfrak F_h$$-normal subgroups of finite groups. (English) Zbl 1226.20011
Only finite groups are considered. Let $$G$$ be a group and let $$\mathfrak F$$ be a formation. A subgroup $$H$$ of $$G$$ is said to be $$\mathfrak F_h$$-normal in $$G$$ if there exists a normal subgroup $$T$$ of $$G$$ such that $$HT$$ is a normal Hall 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 $$G/H_G$$. 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 an extension of the so-called $$\mathfrak F_n$$-supplemented subgroups considered by N. Yang and the second author [in Asian-Eur. J. Math. 1, No. 4, 619-629 (2008; Zbl 1176.20018)]. (Notice that A. Y. Alsheik Ahmad, J. J. Jaraden and A. N. Skiba called them $$\mathfrak U_{\text{c}}$$-normal subgroups, for the class $$\mathfrak U$$ of supersoluble groups, [in Algebra Colloq. 14, No. 1, 25-36 (2007; Zbl 1126.20012)].)
In the paper under review, the authors obtain some results about the influence of $$\mathfrak F_h$$-normality on the structure of groups.
As a sample, they prove: Theorem 3.1. Let $$\mathfrak F$$ be a saturated formation containing $$\mathfrak U$$, the class of supersoluble groups. Then $$G\in\mathfrak F$$ if and only if there exists a soluble normal subgroup $$H$$of $$G$$ such that $$G/H\in\mathfrak F$$ and every maximal subgroup of every Sylow subgroup of $$F(H)$$ not having a supersoluble supplement in $$G$$ is $$\mathfrak U_h$$-normal in $$G$$.
Some characterizations of supersoluble groups, soluble groups and $$p$$-nilpotent groups are also obtained by using this subgroup embedding property.

##### MSC:
 20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, $$\pi$$-length, ranks 20D15 Finite nilpotent groups, $$p$$-groups 20D20 Sylow subgroups, Sylow properties, $$\pi$$-groups, $$\pi$$-structure 20D30 Series and lattices of subgroups 20D40 Products of subgroups of abstract finite groups
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##### References:
 [1] Alsheik A Y, Jaraden J J, Skiba A N. On U c-normal subgroups of finite groups. Algebra Colloquium, 2007, 14: 25–36 · Zbl 1126.20012 [2] Guo W. The Theory of Classes of Groups. Beijing-New York-Dordrecht-Boston-London: Science Press/Kluwer Academic Publishers, 2000 · Zbl 1005.20016 [3] Guo W. On $$\mathfrak{F}$$ -supplemented subgroups of finite group. Manuscripta Math, 2008, 127: 139–150 · Zbl 1172.20019 [4] Guo W, Shum K P, Skiba A N. G-covering subgroups systems for the classes of supersoluble and nilpotent groups. Israel J Math, 2003, 138: 125–138 · Zbl 1050.20009 [5] Guo X, Shum K P. Cover-avoidance properties and the structure of finite groups. J Pure Appl Algebra, 2003, 181: 297–308 · Zbl 1028.20014 [6] Guo X, Shum K P. On c-normal maximal and minimal subgroups of Sylow p-subgroups of finite groups. Arch Math, 2003, 80: 561–569 · Zbl 1050.20010 [7] Huppert B. Endliche Gruppen I. Berlin-Heidelberg-New York: Springer-Verlag, 1967 · Zbl 0217.07201 [8] Li D, Guo X. The influence of c-normality of subgroups on the structure of finite groups II. Comm Algebra, 1998, 26: 1913–1922 · Zbl 0907.20052 [9] Li D, Guo X. The influence of c-normality of subgroups on the structure of finite groups. J Pure Appl Algebra. 2000, 150: 53–60 · Zbl 0967.20011 [10] Miao L, Guo W. On c-supplemented primary subgroups of finite groups. Proc of F Scorina Gomel State Univ, 2004, 6(27): 3–10 [11] Ramadan M. Influence of normality on maximal subgroups of Sylow subgroups of a finite group. Acta Math Hungar, 1992, 59: 107–110 · Zbl 0802.20019 [12] Robinson D J S. A Course in the Theory of Groups. New York: Springer, 1982 · Zbl 0483.20001 [13] Shemetkov L A. Formations of Finite Groups. Moscow: Nauka, 1978 · Zbl 0496.20014 [14] Shemetkov L A, Skiba A N. Formations of Algebraic Systems. Moscow: Nauka, 1989 · Zbl 0667.08001 [15] Wang Y. c-Normality of groups and its properties. J Algebra, 1996, 180: 954–965 · Zbl 0847.20010 [16] Wei H. On c-normal maximal and minimal subgroups of Sylow subgroups of finite groups. Comm Algebra, 2001, 29: 2193–2200 · Zbl 0990.20012 [17] Yang N, Guo W. On $$\mathfrak{F}_n$$ -supplemented subgroups of finite groups. Asian-European J of Math, 2008, 1(4): 619–629 · Zbl 1176.20018
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