# Primary invariants

This question is related to the earlier question which is in the given link: Primary invariants of a finite group

Let $G$ be a finite group and $V$ a complex representation of degree $n$, and let $f_1,f_2,\ldots,f_n \in Sym(V)^G$ be algebraically independent invariants.

Suppose the $f_i$'s satisfy the condition that $\frac{\prod deg(f_i)}{|G|}$ is an integer and also least possible.

Does that mean $f_i$'s are a homogeneous system of parameters?

Here's a counter example: Let the cyclic group of order 2 act on $K[x_1,x_2]$ by $x_i \mapsto -x_i$. Then $f_1 = x_1^2$ and $f_2 = x_1 x_2$ satisfy your hypotheses, but are not primary invariants.
If the quotient you consider is 1, then one can indeed conclude that the $f_i$ are primaries. See G. Kemper, Calculating Invariant Rings of Finite Groups over Arbitrary Fields, J. Symbolic Computation 21 (1996), 351-366.