On The Properties Of Strong Flatness And Flatness Of Modular Representations For Certain Noncyclic 2-groups

The structural complexity of an invariant ring can be measured by its depth.In the reference[10],Ellingsrud and Skjelbred provided a lower bound for the depth of modular invariant rings.If the depth of an invariant ring attains this lower bound,then this representation is called flat.In this paper,we consider the properties of strong flatness,flatness of modular repre-sentations for certain noncyclic 2-groups over fields with characteristic 2,and determine the depth of corresponding modular invariant rings.Particularly,in the indecomposable representations of a noncyclic abelian 2-group of order more than four,there are two types of flat representations.And for non-abelian 2-groups,if the group G is the dihedral group or the generalized quaternion group of order 2n+2,then its faithful representations in dimension 1 + 2n are flat,but not strongly flat.There are three chapters in this paper.The first chapter introduces some basic concepts and relevant theorems that will be useful later.In chapter 2,we generalize the results of Klein four-group to a noncyclic abelian 2-group of order more than four,and obtain that it has two types of flat representations.In chapter 3,we show the flatness of modular representations for non-abelian 2-groups.