On P𝔉-Supplemented subgroups of finite groups

ABSTRACTLet 𝔉 be a class of groups and G a finite group. A maximal subgroup M of G is called 𝔉-abnormal provided G∕MG∉𝔉. Let Ksubgroup of G. Then we say that (K,H) is an 𝔉-abnormal pair of G provided K is a maximal 𝔉-abnormal subgroup of H. Let A be a subgroup of G. Then we say that A is 𝔉-quasipermutable in G provided A either covers or avoids every 𝔉-abnormal pair of G. In this paper, we consider some applications of 𝔉-quasipermutable subgroups.