The Classification Of Z₄-invariant Function Germs,Under The Left-right Equivalence

In this thesis,With the compact Lie group Z₄ as a symmetric group,we dis-cusses the classification and recognition of Z_4^-invariant function germs under the left-right equivalence group action.and give the Corresponding normalized form.This paper consists of five chapters.Introduction:We have made a brief summary of development trends and re-search meaning,the research questions and significance of the paper.In the first chapter:We introduce some basic concepts and some basic conclu-sion.In the second chapter:We mainly give the Hilbert basis of D_4^-invariant function germs and Z_4^-invariant function germs and the relationship between D_4^-invariant function germs ring and Z_4^-invariant function germs ring.In the third chapter:This paper give the D_4^-equivariant mapping germs mod-ule and Z_4^-equivariant mapping germs module as the generator of ?D40-module.In the fourth chapter:We mainly give the unipotent tangent space and tangent space under the action of group???Z₄ and Z_4^-invariant function germs with topo-logical codimension no more than 3 under the action of group???Z₄ are classified,and the corresponding normalized form are obtained.