Very Exceptional Group

The conception of very exceptional group is introduced when we consider Galois extension of a field and it probably reflect the relation between ideal class groups and K-groups of algebraic number fields. Finally a solvable group is very exceptional if and only if different subgroups of prime orders can change and at least one is unique. The main topic of this thesis includes what properties of groups are if they are very exceptional and in which conditions groups are very exceptional, taking Abelian group nilpotent group, solvable group as examples. Chapter one introduces the basic concepts of the topics. Chapter two describes the main result and proofs.