Subnormalizers and embedding properties of subgroups of finite groups

Three important problems concerned with the arrangement of intermediate subgroups are solved. All of them are related to subgroup embedding properties like pronormality. Firstly, it is shown that the subnormalizer condition is equivalent to weak normality for subgroups of a finite supersolvable group. A counterexample to the similar statement in a finite solvable group is constructed. Secondly, we find necessary and sufficient conditions for coincidence of paranormality, pronormality and abnormality with their weak analogs. The important tool for solving such problems is elaborated, it is based on Burnside tables of marks. We found out a counterexample to the long-standing conjecture of Z. I. Borevich on equivalence of polynormality and paranormality in solvable groups. The third part of the paper deals with nilpotent polynormal subgroups of a finite group. A necessary and sufficient condition of polynormality for the solvable case is given. It is shown that the solvability assumption cannot be omitted.