Some New Applications of Weakly ℋ-Embedded Subgroups of Finite Groups

A subgroup H of a finite group G is said to be weakly H -embedded in G if there exists a normal subgroup T of G such that H G = H T and H ∩ T ∈ H ( G ) , where H G is the normal closure of H in G , and H ( G ) is the set of all H -subgroups of G . In the recent research, Asaad, Ramadan and Heliel gave new characterization of p -nilpotent: Let p be the smallest prime dividing | G | , and P a non-cyclic Sylow p-subgroup of G. Then G is p-nilpotent if and only if there exists a p-power d with 1 < d < | P | such that all subgroups of P of order d and p d are weakly H -embedded in G . As new applications of weakly H -embedded subgroups, in this paper, (1) we generalize this result for general prime p and get a new criterion for p -supersolubility; (2) adding the condition “ N G ( P ) is p -nilpotent”, here N G ( P ) = { g ∈ G | P g = P } is the normalizer of P in G , we obtain p -nilpotence for general prime p . Moreover, our tool is the weakly H -embedded subgroup. However, instead of the normality of H G = H T , we just need H T is S -quasinormal in G , which means that H T permutes with every Sylow subgroup of G .