= Solution
A <semisimple module> is a direct sum of <simple modules>. A finite-dimensional <semisimple algebra> is one whose left regular <module> is semisimple; equivalently, all its finite-dimensional <modules> are semisimple, or every <submodule> of a finite-dimensional <module> has a complementary <submodule>.
The <Maschke theorem> states that, for a finite <group> $G$ and a <field> $k$ with $|G|\ne0$ in $k$, every finite-dimensional $kG$-<module> is semisimple, so $kG$ is a <semisimple algebra>. The assumption is equivalent to saying that the <characteristic of a field> does not divide $|G|$. The averaging construction makes a projection onto a <submodule> equivariant by replacing $P$ with $|G|^{-1}\sum_{g\in G}gPg^{-1}$.
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