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A note on a result of Guo and Isaacs about \(p\)-supersolubility of finite groups. (English) Zbl 1342.20020

Summary: In this note, global information about a finite group is obtained by assuming that certain subgroups of some given order are S-semipermutable. Recall that a subgroup \(H\) of a finite group \(G\) is said to be S-semipermutable if \(H\) permutes with all Sylow subgroups of \(G\) of order coprime to \(|H|\). We prove that for a fixed prime \(p\), a given Sylow \(p\)-subgroup \(P\) of a finite group \(G\), and a power \(d\) of \(p\) dividing \(|G|\) such that \(1\leq d<|P|\), if \(H\cap O^p(G)\) is S-semipermutable in \(O^p(G)\) for all normal subgroups \(H\) of \(P\) with \(|H|=d\), then either \(G\) is \(p\)-supersoluble or else \(|P\cap O^p(G)|>d\). This extends the main result of Y. Guo and I. M. Isaacs [in Arch. Math. 105, No. 3, 215-222 (2015; Zbl 1344.20032)]. We derive some theorems that extend some known results concerning S-semipermutable subgroups.

MSC:

20D40 Products of subgroups of abstract finite groups
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure

Citations:

Zbl 1344.20032
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References:

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