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?



A question about an application of Molien's formula to find the generators and relations of an invariant ring
As pointed out by Richard Stanley, there are some subtleties in finding the invariant generators. 


The Molien series only tells you the Hilbert series of the ring of invariants up to *reduced* rational forms. So some information is lost, as illustrated in Richard's answer. 


One can gain some information by looking at the actual series and a bit of work. For example, if your Molien series looks like:


$$1 + 2t^2 + 4t^4 +...$$ 


Then you know there are two invariant forms of degrees $2$. They generate *three* elements in degree $4$ ($f^2,fg,g^2$) so there must be another algebraically independent form in degree $4$, etc...


Once you know the degree of the generators, a decent way to guess them is to use the Reynolds operator (I assume the group is finite of order prime to characteristic of the field): 


$$f \mapsto 1/|G|\sum\_{g\in G} gf$$ 


Starting with any polynomial $f$, this always lands in the invariant subring, as you can easily check. 


Once you know the first few generators, try to match the Hilbert series of the subalgebra generated by those with your Hilbert series. If they don't match, there must be another generator of degree equals to the first point the series mismatched, and on you go...


ADDED: I originally tried not to mention the word "Cohen-Macaulay", but it is particularly relevant here, so perhaps I should. 


The Reynolds operator mentioned above give you a splitting $R^G \to R= \mathbb C[x\_1,\cdots, x\_n]$. Such splitting implies that $R^G$ is Cohen-Macaulay. 


Why is that relevant? The point is that once you come up with $n$ invariant forms, say $f\_1,\cdots f\_n$ of degrees $d\_1,\cdots d\_n$, which you happen to know is a regular sequence on $R$ (hence also regular on $R^G$). Then as $R^G$ is Cohen-Macaulay, it is *free* over $R' = \mathbb C[f\_1, \cdots, f\_n]$. Assuming you can write 


$$R^G = \oplus\_1^m h\_jR'$$ 


with each $h\_j$ of degree $e\_j$, then the Hilbert series of $R^G$ must be 


$$\frac{\sum t^{e\_j}}{\prod(1-t^{d\_i})} $$


So by comparing with the Molien series you know the degree of each $h\_j$. (See this [survey by Richard Stanley](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.152.2939)!) 


