Conditions on p-subgroups implying p-nilpotence or p-supersolvability

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

“…We mention that Corollary C and Corollary E of [2] by Guo and Isaacs are immediate consequences of Theorems 4 and 5. Note that the proof of Corollary E in [2] is incomplete.…”

“…An interesting idea of [2] is that in the hypothesis of the theorem, only the normal subgroups of order d are considered, not necessarily the set of all subgroups of order d, at the drawback of obtaining as a conclusion either p-supersolubility or a restriction on the order of the Sylow p-subgroup of O p (G).…”

A note on the embedding properties of p-subgroups in finite groups

“…It is quite interesting to investigate the structure of a group by using some kind of the embedding properties of subgroups and many interesting results have been given (for example, see [1,6,8,13]). Recently, Guo and Isaacs [6] proved the following theorem by using the embedding condition "H ∩ O p (G) ¢ O p (G)".…”

“…It is quite interesting to investigate the structure of a group by using some kind of the embedding properties of subgroups and many interesting results have been given (for example, see [1,6,8,13]). Recently, Guo and Isaacs [6] proved the following theorem by using the embedding condition "H ∩ O p (G) ¢ O p (G)". An interesting idea of [6] is that in the hypothesis of the theorem only the normal subgroups of order d are considered, not necessarily the family of all subgroups of order d. Recall that a subgroup H of a group G is said to be S-semipermutable in G (see [12]) if H permutes with all Sylow q-subgroups of G for the primes q not dividing |H|.…”

“…Recently, Guo and Isaacs [6] proved the following theorem by using the embedding condition "H ∩ O p (G) ¢ O p (G)". An interesting idea of [6] is that in the hypothesis of the theorem only the normal subgroups of order d are considered, not necessarily the family of all subgroups of order d. Recall that a subgroup H of a group G is said to be S-semipermutable in G (see [12]) if H permutes with all Sylow q-subgroups of G for the primes q not dividing |H|. Ballester-Bolinches etc in their paper [1] proved an analogous result, but they only assume that H ∩ O p (G) are S-semipermutable in O p (G) instead of assuming that these subgroups are normal in O p (G).…”