p -Hypercyclically embedding and Π-property of subgroups of finite groups.

LetGbe a finite group,Ea normal subgroup ofGandpa fixed prime. We say thatEisp-hypercyclically embeddedinGif everyp-G-chief factor ofEis cyclic. A subgroupHofGis said to satisfy Π-propertyinGif |G∕K:NG∕K((H∩L)K∕K)| is aπ((H∩L)K∕K)-number for any chief factorL∕KinG; we say thatHhas Π -property inGifH∩Oπ(H)(G) has Π-property inG. In this paper, we prove thatEisp-hypercyclically embedded inGif and only if some classes ofp-subgroups ofEhave Π -property inG. Some recent results are extended. [ABSTRACT FROM AUTHOR]