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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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