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Primary subgroups and the structure of finite groups. (English) Zbl 1506.20013

Summary: Let \(G\) be a group and \(H \le G.\) The permutizer of \(H\) in \(G\) is \(P_G(H) = \langle x \mid x \in G, \langle x \rangle H = H \langle x\rangle\rangle.\) \(H\) is said to be strongly permuteral in \(G\) if \(P_U(H) = U\) whenever \(H \le U \le G.\) Moreover, let \(P_{G_P}(H) = \langle x \mid x \text{ is a primary element of } G, \langle x \rangle H = H \langle x \rangle\rangle.\) \(H\) is said to be strongly \(\mathbb P\)-permuteral in \(G\) if \(P_{U_P} (H) = U\) whenever \(H \le U \le G.\) In this article, we study the structure of a group \(G\) in which every Sylow subgroup and its maximal subgroups are strongly \(\mathbb P\)-permuteral or abnormal and the structure of \(G\) with self-normalizing or strongly permuteral Sylow 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
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