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Adolfo Ballester-Bolinches, Ramón Esteban-Romero, and ShouHong Qiao.
A note on a result of Guo and Isaacs about p-supersolubility of finite groups.
Arch. Math. (Basel), 106(6):501–506, 2016.
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≤d<|P|, if H∩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∩O^p(G)|>d. This extends the main result of Guo and Isaacs in (Arch. Math. 105:215–222 2015). We derive some theorems that extend some known results concerning S-semipermutable subgroups.
2010 Mathematical Subject Classification: 20D10, 20D20
Keywords: Finite group, p-supersoluble group, S-semipermutable subgroup