Modular Invariants Of Finite Orthogonal Groups

Suppose p 1s a prime number, r ∈ N+ and Fq is a finite field with q = pr elements. Let SO(3,Fq) and Ca(3,Fq) be the 3-dimensional special and conformal orthogonal group-s over Fq respectively, acting on the polynomial ring Fq[x1,x2,x3] as Fq-automorphism groups. In this paper, we prove that the ring of invariants Fq[x1,x2,x3]SO(3,Fq) is a hyper-surface, i.e., we find 4 invariants Q30,Q31,Q2*,Q4 ∈Fq[x1,x2,x3]SO(3,Fq)such that Fq[x1,x2,x3]SO(3,Fq)=Fq[Q30,Q31,Q2*,Q4].We also find a generating set of IFq[x1,x2,x3]CO(3,Fq)and a free basis of Fq[x1,x2,x3]CO(3,Fq)over its HSOP.