On Metabelian Minimal Irregular P-groups

Suppose that p is a prime and G is a finite p-group.We say G is regular if(ab)p=apbpcp1�cpm for all a,b ? G,where ci?<a,b>'.Regular p-groups are a wide and beautiful extension of abelian p-groups.On the other hand,in a sense,the bulk of finite p-groups is regular.Hence the study of regular p-groups is a main content in that of finite p-groups It is a basic method to study regularity by using minimal irregular p-groups.Hence it,is very important to determine the structure of minimal irregular p-groups.However,this is quite difficult and complex.Up to know.the classification of minimal irregular p-groups is only given for p=2 and 3,respectively.By a way,the case of p=2 is trivial.For p>5,the problem of the classification of minimal irregular p-groups is open.The thesis tries to study the problem.We notice that the class of metabelian p-groups is a larger class of p-groups.In this thesis we pay attention on the study of metabelian ininiinal irregular p-groups.The main results are as follows1.For p>5,minimal irregular p-groups with an abelian maximal subgroup are classified.2.For p=5,the factor groups G/(?)1(G)of metabelian minimal irregular 5-groups G are classified,where the isomorphism problem is solved by Magma.Based on the classification.metabelian minimal irregular 5-groups are also classified.