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Finite groups with systems of \(\Sigma\)-\( \mathfrak{F} \)-embedded subgroups. (English) Zbl 1454.20027

Summary: Let \(\mathfrak{F}\) denote a class of groups. A maximal subgroup \(M\) of \(G\) is called \(\mathfrak{F} \)-abnormal provided \(G/M_G \not\in \mathfrak{F} \). We say that \((K,H)\) is an \(\mathfrak{F} \)-abnormal pair of \(G\) provided \(K\) is a maximal \(\mathfrak{F} \)-abnormal subgroup of \(H\). Let \(\Sigma = \{G_0 \leq G_1 \leq G_2 \leq \dots \leq G_n\}\) be a subgroup series of \(G\). A subgroup \(H\) of \(G\) is said to be \(\Sigma\)-\( \mathfrak{F} \)-embedded in \(G\) if \(H\) either covers or avoids every \(\mathfrak{F} \)-abnormal pair \((K,H)\) such that \(G_{i -1} \leq K < H \leq G_i\) for some \(i \in \{0, 1, \dots, n\}\). In this paper, some new characterizations of \(p\)-supersoluble and \(p\)-soluble are given by discussing the properties of \(\Sigma\)-\( \mathfrak{F} \)-embedded of subgroups.

MSC:

20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20D30 Series and lattices of subgroups
20D35 Subnormal subgroups of abstract finite groups
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References:

[1] Guo, W., The theory of classes of groups (2000), Beijing-New York-Dordrecht-Boston-London: Science Press-Kluwer Academic Publishers, Beijing-New York-Dordrecht-Boston-London · Zbl 1005.20016
[2] Doerk, K.; Hawkes, T., Finite soluble groups (1992), Berlin-New York: Walter de Gruyter, Berlin-New York · Zbl 0753.20001
[3] W. Guo, Structure theory for cononical classes of finite groups, Springer, 2015. · Zbl 1343.20021
[4] Guo, W.; Skiba, A. N., Finite groups with systems of Σ-embedded subgroups, Science China Mathematics, 54, 1909-1926 (2011) · Zbl 1255.20018 · doi:10.1007/s11425-011-4270-1
[5] Kegel, O. H., Untergruppenverbande endlicher Gruppen, die den subnormalteilerverband each enthalten, Arch. Math. (Basel), 30, 225-228 (1978) · Zbl 0943.20500 · doi:10.1007/BF01226043
[6] Ballester-Bolinches, A.; Ezquerro, L. M., Classes of finite groups (2006), Dordrecht: Springer, Dordrecht · Zbl 1102.20016
[7] Ballester-Bolinches, A.; Esteban-Romero, R.; Asaad, M., Products of finite groups (2010), Berlin-New York: Walter de Gruyter, Berlin-New York · Zbl 1206.20019
[8] Guo, W.; Skiba, A. N., On \(\mathfrak{F}\) Φ*-hypercentral subgroups of finite groups, J. Algebra, 372, 275-292 (2012) · Zbl 1283.20010 · doi:10.1016/j.jalgebra.2012.08.027
[9] Huppert, B., Endliche Gruppen I (1967), Berlin: Springer, Berlin · Zbl 0217.07201
[10] D. Gorenstein, Finite groups, New York-Evanston-London, 1968. · Zbl 0185.05701
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