On nearly S-permutably embedded subgroups of finite groups

Let Gbe a finite group. A subgroup H of G is said to be S-permutable in G if HP = PH for all Sylow subgroups P of G. A subgroup A of a group G is said to be S-permutably embedded in G if for each Sylow subgroup of A is also a Sylow of some S-permutable subgroup of G. In this paper, we analyze the following generalization of this concept. Let H be a subgroup of a group G. Then we say that H is nearly S-permutably embedded in G if G has a subgroup T and an S-permutably embedded subgroup C a parts per thousand currency sign H such that HT = G and T a (c) H a parts per thousand currency sign C. We study the structure of G under the assumption that some subgroups of G are nearly S-permutably embedded in G. Some known results are generalized.