nLab Maschke's theorem

Maschke’s theorem

Given a finite group $G$ and a field $k$, the group algebra $k[G]$ is semisimple iff the characteristic of $k$ does not divide order of the group (that is, if $|G|$ is an invertible number in $k$, which is always in characteristic zero).

If the base $k$ is just a commutative unital ring, then there is the following statement:

Theorem

If $|G|1_k$ is invertible in $k$, then an exact sequence of $k[G]$-modules splits iff it splits after applying the forgetful functor from $k[G]$-modules to $k$-modules (and the splitting in ${}_{k[G]}Mod$ can be functorially constructed from the splitting in ${}_{k}Mod$).

If $k$ is a field, it follows that the $k[G]$ is semisimple, so this statement can be understood as a generalization of Maschke’s theorem. This is also one of the motivations for the concept of a separable functor.

The importance of the classical Maschke’s theorem is that much is known about the structure of semisimple ring?s (starting with, e.g., Wedderburn's theorem?).

• wikipedia Maschke’s theorem