Notes on Finite Groups with Nearly S-permutable and Nearly S-permutable-Transitive Subgroups

Let G be a finite group and let H be a subgroup of G. We called H is nearly S-permutable in G if for every prime p such that (p, |H|) =1 and every subgroup K of G containing H the normalizer NK(H) contains some Sylow p-subgroup of K. We shall denoted this by (H is NSP in G). We introduce the class of NSPT-groups as those groups in which nearly S-permutability is transitive among subgroups. Specifically, if A is NSP in B and B is NSP in G, then A is NSP in G. In this paper we study certain finite groups characterized by NSP and NSPT properties and we also compare specific subgroups with groups under study, supported by theorems and examples.