If $\mathrm{char}k\nmid \mathrm{\mid }G\mathrm{\mid }$, then $k[G]$ is a semisimple ring

Given: A finite group $G$ and a field $k$ with $\mathrm{char}k\nmid \mathrm{\mid }G\mathrm{\mid }$.

If $G$ is finite, then ${\dim }_{k}k[G]=\mathrm{\mid }G\mathrm{\mid }$ (If $G$ is finite then ${\dim }_{k}k[G]=\mathrm{\mid }G\mathrm{\mid }$).

The regular representation of $G$ over $k$ is the action on $k[G]$ by left multiplication, namely $g\cdot x=[g]x$ (The trivial representation, the regular representation, and permutation representations from finite $G$-sets).

Under the characteristic hypothesis, every finite-dimensional representation of $G$ over $k$ is completely reducible (If $\mathrm{char}k\nmid \mathrm{\mid }G\mathrm{\mid }$, every finite-dimensional representation of $G$ is completely reducible).

A representation is completely reducible exactly when its underlying space is an internal direct sum of irreducible subrepresentations (A completely reducible representation as a finite direct sum of irreducible subrepresentations).

Under the dictionary, irreducible representations are exactly simple left $k[G]$-modules (Under the dictionary, subrepresentations are exactly submodules and irreducible representations are exactly simple modules).

A unital ring is semisimple exactly when its left regular module is semisimple (A semisimple ring as a ring whose left regular module is semisimple).