be a finite group, and let $R$ be a left $G$-module algebra. Set $A=R^G:=\{a\in R|g\cdot a=a,\forall\  g\in G\}$. If $R$ is a free $A$-module with a free basis $y_0=1,y_1,\dots,y_n\in R$ satisfying the conditions:

[(i)]{} $g(y_1,\dots,y_n)=(\xi_1y_1,\dots,\xi_ny_n)$ where $\xi_i$ is a $j_i$-th ($j_i\ge3$) primitive root of unity for all $i=1,\dots,n$,

[(ii)]{} $y_iy_j=0$ for $i,j=1,\dots,n$,\
then $R$ is $G$-semisimple.

By Lemma \[lem-pertinred\], we may assume that pertinent sequences have the form $(a_0,a_1,\dots,a_n)\overset{G}\sim(1,y_1,\dots,y_n)$. By the condition (ii), we may assume $a_1,\dots,a_n\in A$. Assume $a_0=b_0+b_1y_1+\cdots+b_ny_n$ with $b_i\in A$ for $i=0,\dots,n$. By the equality $\sum_{i=0}^na_ig(y_i)=0$ we obtain $b_0=0$ and $b_i=-\xi_ia_i$ for $i=1,\dots,n$. By the equality $\sum_{i=0}^na_ig^2(y_i)=0$, we obtain $b_i=-\xi_i^2a_i$ for $i=1,\dots,n$. Since $\xi_i$ is a $j_i$-th primitive root of unity with $j_i\ge3$ for $i=1,\dots,n$, we have $b_i=a_i=0$ for $i=1,\dots,n$. Hence $\mfr(R,G)=0$.

Group actions with $\Pty(R,G)=1$ {#sec3}
================================

Recall that we have a definition of the pertinency of a group action on an algebra in [@BHZ], where we had to assume that the group is finite. In order to avoid this restriction, we provide an alternative (equivalent) definition in this paper. We will see, in some cases, the definition here is more flexible.

Let $G$ be group, and let $R$ be a noetherian algebra with finite Gelfand-Killilov 