A note on a result of Guo and Isaacs about p-supersolubility of finite groups

Abstract

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 ${\lvert H\rvert}$. 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 ${\lvert G\rvert}$ such that ${1\le d < \lvert P\rvert}$, if ${H\,{\cap}\, O^p(G)}$ is S-semipermutable in ${O^p(G)}$ for all normal subgroups H of P with ${\lvert H\rvert=d}$, then either G is p-supersoluble or else ${\lvert P\,{\cap}\, {O^p(G)}\rvert > d}$. 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.