(GK) dimension. The [*pertinency*]{} of the $G$-action on $R$ is defined to be the number $$\label{pty1}
  \Pty(R,G)=\operatorname{GKdim}(R)-\operatorname{GKdim}(\mathfrak{R}(R,G)).$$

Let $R$ be a noetherian $G$-module algebra with finite GK-dimension, and let $B:=R*G$ be the skew group algebra. Assume $G$ is finite. Let $\kk G$ be the group algebra. As before, let $e=\frac{1}{|G|}\sum_{g\in G}g$. Recall from [@BHZ] that the pertinency of the $G$-action is defined to be $$\label{pty2}
  \Pty(R,G)=\operatorname{GKdim}(R)-\operatorname{GKdim}(B/(BeB)).$$

\[lem1\] Assume $G$ is a finite group. Let $R$ be a noetherian left $G$-module algebra with finite GK-dimension. Then $\operatorname{GKdim}(B/(BeB))=\operatorname{GKdim}(\mathfrak{R}(R,G))$.

As before, let $\iota:R\to B,a\mapsto a\otimes 1$ be the inclusion map. Let $\pi:B\to B/BeB$ be the projection map. Then $\overline{R}:=\pi\circ\iota(R)$ is a subalgebra of $B/BeB$. Since $B$ is finitely generated as a left $R$-module, we see that $B/BeB$ is a finitely generated $\overline{R}$-module. By [@KL Proposition 5.5]$, \operatorname{GKdim}(\overline{R})=\operatorname{GKdim}(B/BeB)$. As in the proof of Proposition \[prop-mrad\], we see $\mfr(R,G)=R\cap BeB$. Therefore $\overline{R}\cong \mathfrak{R}(R,G)$.

As a consequence of Lemma \[lem1\], we obtain:

With the assumptions in Lemma \[lem1\], the pertinencies defined by (\[pty1\]) and (\[pty2\]) are equivalent.

In order to compute the pertinency of a group action, one has to analyze the radical ideal of the group action. Next we provide several ways to find elements 