In this thesis, we are interested in ring of invariant. Classical invariant theory was a hot topic in the 19th century and in the beginning of the 20th century. We study polynomials which remain invariant under the action of finite matrix group G. The result is a collection of algorithms for finding a finite set {I1, …,In} of fundamental invariants which generate the invariant subring . We introduce Molien series in section 4, to aid in the calculation of invariant subring and introduce the Choen-Macaulay properties in section 5. In section 6, we prove that the ring of generalized symmetric polynomials is Choen-Macaulay over . The most important result lies in section 7. When m=2, we find an explicit basis of ring of generalized symmetric polynomials over .