property of $G$-semisimple $G$-module algebras.

\[prop-subalg\] Let $R$ be a left $G$-semisimple $G$-module algebra. If $S$ is a subalgebra of $R$ closed under the $G$-action, then $S$, as a $G$-module algebra, is $G$-semisimple.

Pertinency algebras {#sec2}
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Let $G$ be a group, and let $R$ be a $G$-module algebra. We will review some basic properties of the $G$-radical of $R$, which justify the name “radical”.

Let $R$ be a $G$-module algebra with $G$-action $\theta:G\to \operatorname{Aut}(R)$.

-   The $G$-radical $\mfr(R,G)$ is a $G$-ideal, i.e., it is closed under the $G$-action.

-   The $G$-action $\theta$ induces a $G$-action $\overline{\theta}:G\to \operatorname{Aut}(\mathfrak{R}(R,G))$ so that $\mathfrak{R}(R,G)$ is also a left $G$-module algebra.

-   If $G$ is a finite group, then $\mfr(\mathfrak{R}(R,G),G)=0$, i.e., the $G$-module algebra $\mathfrak{R}(R,G)$ is $G$-semisimple.

The statement (i) follows from Proposition \[prop-basic\](i). Then statement (ii) follows from (i).

\(iii) For an element $a\in R$, we write $\overline{a}$ for the image of $a$ in $\mathfrak{R}(R,G)$. Let $(a_1,\dots,a_n)$ and $(b_1,\dots,b_n)$ be sequences of elements in $R$ such that $(\overline{a}_1,\dots,\overline{a}_n)$ and $(\overline{b}_1,\dots,\overline{b}_n)$ are pertinent under the $G$-action in $\mathfrak{R}(R,G)$. Since $G$ is a finite group, we assume $|G|=k$ and we may label the elements of $G$ in $\{1=g_0,g_1,\dots,g_{k-1}\}$. Then we have $\sum_{i=1}^na_i(g_j\cdot b_i)\in \mfr(R,G)$ for 