Let a finite group \(G\) act linearly on \(\CC^n\) through a representation \(\rho\), and so on \(\CC[x_1,\ldots,x_n]\). The invariant ring \(\CC[x_1,\ldots,x_n]^G\) is finitely generated, and the Reynolds operator \(f\mapsto\frac{1}{\abs{G}}\sum_{g\in G}g\cdot f\) is a projection onto it; by Noether’s bound, its values on the monomials of degree at most \(\abs{G}\) generate the invariant ring, and a Gröbner basis computation finds the relations among generators. Molien’s formula expresses the Hilbert series through \(\rho\): \(\sum_{d\ge0}\dim_\CC\big(\CC[x_1,\ldots,x_n]^G\big)_d\,t^d=\frac{1}{\abs{G}}\sum_{g\in G}\frac{1}{\det(I-t\rho(g))}\). The Hilbert series depends on \(\rho\) and not only on the abstract group: the subgroups \(\{\pm I\}\) and \(\{I,\operatorname{diag}(1,-1)\}\) of \(\GL_2(\CC)\) are both cyclic of order \(2\), with Hilbert series \(\frac{1+t^2}{(1-t^2)^2}\) and \(\frac{1}{(1-t)(1-t^2)}\).