which there are tow equal rows, and hence it is zero. Therefore, $\delta_G(a_1,\dots,a_n)=\sum_{i=1}^na_iA_i\in \mfr(R,G)$.

\[prop2-2\] Let $R$ be a noetherian algebra which is a domain with $1\leq \operatorname{GKdim}(R)<\infty$, and let $G\subseteq \operatorname{Aut}(R)$ be a finite subgroup with $|G|\neq 1$ and $\operatorname{char}\kk\nmid |G|$. If $R$ and $G$ satisfy one of the following conditions, then $\Pty(R,G)\ge1$.

-   $R$ is connected graded, the $G$-action preserves the grading of $R$, and the order of $G$ is prime.

-   $G$ acts faithfully on the center of $R$.

-   $R=\kk_{q_{ij}}[x_1,\dots,x_n]$, $|G|\leq n+1$ and every $g\in G$ acts on $x_1,\dots,x_{n}$ diagonally.

\(i) If $|G|$ is a prime number, then $G$ is a cyclic group. Assume $G=\langle g\rangle$. Since $R$ is noetherian, $\dim R_i<\infty$ for all $i\ge0$. Since the order of $g$ is finite, we may choose a basis $S$ of $R_1$ so that $g$ acts on the basis $S$ diagonally. Thus for any element $a\in S$, $g(a)=\xi a$ for some $p$-th root of unit. Letting all the elements $a_i=a$ in Lemma \[lem2\], we see $a^p\in \mfr(R,G)$. Since $R$ is a domain, $a^p\neq 0$. It follows that $\operatorname{GKdim}(R/\mfr(R,G))\leq \operatorname{GKdim}(R/Ra^pR)\leq \operatorname{GKdim}(R)-1$. Hence $\Pty(R,G)\ge1$.

\(ii) Let $Z(R)$ be the center of $R$. Since $G$ acts on $Z(R)$ faithfully, for any 