| 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. |
