On Π-property and Π-normality of subgroups of finite groups

Let H be a subgroup of group G. H is said to satisfy Π-property in G, if |G/K:NG/K(HK/K∩L/K)| is a π(HK/K∩L/K)-number for any chief factor L/K of G, and, if there is a subnormal supplement T of H in G such that H∩T⩽I⩽H for some subgroup I satisfying Π-property in G, then H is called Π-normal in G. These properties are common properties satisfied by many subgroups which satisfy some known embedding property. Groups can be described when some primary subgroups are Π-normal, and many known results are generalized.