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Sufficient conditions for supersolubility of finite groups. (English) Zbl 0928.20020
A subgroup \(H\) of a finite group \(G\) is \(S\)-quasinormal in \(G\) if it permutes with each Sylow subgroup of \(G\); a subgroup \(K\) of \(G\) is \(S\)-quasinormally embedded in \(G\) if each Sylow subgroup of \(K\) is also a Sylow subgroup of some \(S\)-quasinormal subgroup of \(G\). The authors prove: Theorem 1. If each maximal subgroup of the Sylow subgroups of \(G\) is \(S\)-quasinormally embedded in \(G\), then \(G\) is supersolvable. Theorem 2. If \(G\) is a solvable group with normal subgroup \(H\) such that \(G/H\) is supersolvable and all maximal subgroups of the Sylow subgroups of the Fitting subgroup of \(H\) are \(S\)-quasinormally embedded in \(G\), then \(G\) is supersolvable.

MSC:
20D40 Products of subgroups of abstract finite groups
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20E28 Maximal subgroups
20D25 Special subgroups (Frattini, Fitting, etc.)
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
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[1] Asaad, M.; Ramadan, M.; Shaalan, A., Influence of π-quasinormality on maximal subgroups of Sylow subgroups of Fitting subgroup of a finite group, Arch. math., 56, 521-527, (1991) · Zbl 0738.20026
[2] Ballester-Bolinches, A., Permutably embedded subgroups of finite soluble groups, Arch. math., 65, 1-7, (1995) · Zbl 0823.20020
[3] Deskins, W.E., On quasinormal subgroups of finite groups, Math. Z., 82, 125-132, (1963) · Zbl 0114.02004
[4] Doerk, K.; Hawkes, T., Finite soluble groups, (1992), Walter De Gruyter Berlin · Zbl 0753.20001
[5] Huppert, B., Endliche gruppen I, (1983), Springer Berlin
[6] Kegel, O.H., Sylow-gruppen und subnormalteiler endlicher gruppen, Math. Z., 78, 205-221, (1962) · Zbl 0102.26802
[7] Srinivasan, S., Two sufficient conditions for the supersolvability of finite groups, Israel J. math., 35, 210-214, (1980) · Zbl 0437.20012
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