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Subnormal subgroups and \(p\)-super solvability of finite groups. (Chinese. English summary) Zbl 1363.20022

Summary: Assume that group a \(G\) has a solvable normal subgroup \(H\) and \(G/H\) is a \(p\)-super solvable group. We prove that: (1) If the maximal subgroup of Sylow \(p\)-subgroup \(P\) of \(H\) is subnormal in \(G\), then \(G\) is a \(p\)-super solvable group; (2) If the maximal subgroup of \(O_{p'}(H)\) is included in \(F_p(H)\) and is subnormal in \(G\), then \(G\) is a \(p\)-super solvable group.

MSC:

20D35 Subnormal subgroups of abstract finite groups
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
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