In the very beginning of the book "Introduction to Invariants and Moduli" Shigeru Mukai proves Molien's formula for the Hilbert series of the invariant ring of a finite group action on $\mathbb C^n$. For example, in the case of the standard action of Quaternions on $\mathbb C^2$ the Hilbert series is $\frac{1-t^{12}}{(1-t^4)^2(1-t^6)}$.

After this Mukai explains that this formula hints us that the ring of invariants can be generated by two elements of order $4$ and one element of order $6$, and indeed such elements can be found: $A=x^4+y^4$, $B=x^2y^2$, and $C=xy(x^4-y^4)$. Then one finds a relation $C^2=A^2B -4B^3$ and this gives a complete description of the ring of invariants.

Moreover the same procedure is shown to work in several other cases (e.g. binary icosahedral group).

**My question is a follows:** Is there some theorem that say that this heuristics works often? Namely, if we have an action of a finite group $G$ on $\mathbb C^n$, in order to describe the ring of invariants, we first look on the denominator of the Hilbert series (given by Molien's formula) and try to associate an invariant polynomial of degree $n$ to each factor $(1-t^n)$ (so that this gives us a full set of generators). Or at least, in practice, is this the first thing that one tries to do?