$1\neq g\in G$, there is element $a\in Z(R)$ such that $g(a)\neq a$. Since $R$ is a domain, Proposition \[prop2-1\] asserts that there is a nonzero element $b\in\mfr(R,G)$. The rest of the proof is similar to the case (i).

\(iii) Similar to the above case, the assertion follows from Proposition \[prop2-1\].

The above result shows that in reasonable cases the pertinency of a finite group action is nonzero, which implies the following primeness of the skew group algebra.

\[prop2-3\] Let $R$ be a neotherian locally finite graded algebra which is a domain with $1\leq \operatorname{GKdim}(R)<\infty$. Let $G$ be a nontrivial finite subgroup of $\operatorname{Aut}(R)$ such that the restriction of $G$ to the center of $R$ is faithful. Then the skew group algebra $R*G$ is prime.

By [@BHZ Lemma 3.10], $R*G$ is prime if and only if $\Pty(G,R)\ge1$. Now the result follows from Proposition \[prop2-2\].

The finite group actions of pertinency 1 is of special interest. If $R$ is a polynomial ring, then we have the following version of Shephard-Todd-Chevalley Theorem.

\[prop2-4\] Let $V$ be a finite dimensional vector space, and $R=\kk[V]$ the polynomial algebra. Let $G\subseteq GL(V)$ be a finite subgroup. Then for the invariant subalgebra $R^G$, $\operatorname{gldim}(R^G)<\infty$ if and only 