For a finite group $G$ and complex representation V of degree $n$, I would like to know the precise definition of Primary invariants. Does any set of n algebraically independent homogeneous invariants qualify to be called primary ? Or does the set need to satisfy additional conditions ?
I would like to know specifically the following : For a set $f_1,f_2,\ldots,f_n$ of algebraically independent homogeneous invariants, is the full ring of invariants of G a free module over the subalgebra generated by $f_i$'s.
What I have done:For the group $S_5$ and its irreducible representation of degree 6 (This is not generated by reflections), I have found six homogeneous invariants . I also computed the degrees of them whose product is a multiple of the order of $S_5$.