| This thesis studies the isomorphic problem of subgroups of a,group.Obviously,isomorphic subgroups are of the same order,however,subgroups of the same order are not necessarily isomorphic.A natural question is:what can be said about the group with the property that subgroups of the same order are isomorphic?This question is followed by interests.It should be mentioned that academician Jiping Zhang in his paper[Comm.Algebra,23:5(1995),1605-1612]determined completely the structure of finite groups whose subgroups of the same order are isomorphic.In particular,he classified finite p-groups with the same as property.Related questions to this question are studied by many researchers of group theory.As a continuation of Jiping Zhang’s work and a generalization about the result of p-groups of Jiping Zhang,In 2021,Hongli Zhang,in her thesis,classified finite p-groups whose non-cyclic subgroups of the same order are isomorphic.In this thesis,we,by weakening the condition " non-cyclic subgroups of the same order are isomorphic" to "non-cyclic subgroups of a fixed order are isomorphic",study the structure of such p-groups with this property.Strictly speaking,assume G is a group of order pn.for a fixed k with 1≤k≤n-1,we study the structure of the p-groups whose non-cyclic subgroups of a fixed order are isomorphic.Such p-groups are called BIk-groups.Obviously,every finite p-group is either a BI1-group or a BI2-group.We classify BI3-groups,BI4-groups,and BIk with a cyclic subgroup of order pk.This thesis consists of three parts.The first part is the introduction,which mainly introduces the research background,methods and main results of this thesis.The second part is the preliminary knowledge,which mainly introduces the concepts and known conclusions to be used in this thesis,and proves some basic properties of BIk-groups.In the third part,we classify the BI3-groups,BI4-groups and BIk with a cyclic subgroup of order pk. |
