Finite P-groups With Many Normal Subgroups Or With Unique Subgroup Of Given Structure

Finite p-groups play an important and basic role in the finite group theory.After the classification of finite simple groups is finally completed, the study offinite p-groups becomes more and more active. Many group specialists turn theirattention to the study of finite p-groups, for example, G. Glauberman, Z. Janko,A. Mann and so on.However, the finite p-groups are so complicated that it is quite difcult togive a complete classification of all non-isomorphic p-groups. So to study p-groupsby the normality of the subgroups becomes one of topics in finite groups. Forexample, the finite groups whose subgroups are all normal were described by R.Dedekind and, in the general case, by R. Baer, and so on. In this thesis, we studyp-groups by the normality of the subgroups.In chapter III, we are interested in the structure of J-groups of prime powerorder. Recall that a group G is called a J-group if each x∈G satisfies either<x>ï¿ G, or x, x^gï¿ G for all g∈G\N_G（<x>）. We classify the J-groups of oddprime power order.In chapter IV, we study the C（p^w）-groups. A p-group is called a C（p^w）-groupif the normal closure of every non-normal cyclic subgroup has index at most pw.We prove that the order of a non-Dedekind C（p^w）-group cannot exceed p^4w+4when p>2and G≤H^Gfor every non-normal cyclic subgroup H. We alsocompletely classify non-Dedekind C（p²ï¼‰-groups for p>2.In chapter V, we classify the finite p-groups with exactly one minimal non-abelian subgroup of given structure of order p³.In chapter VI, we classified the finite p-groups whose subgroups of given order are all isomorphic and abelian.