nLab Maschke's theorem

Statement

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.

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 $k[G]$ is semisimple, so that Thm. can be understood as a generalization of Maschke’s theorem . This is also one of the motivations for the concept of separable functors.

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

Related entries

• separable functor

Literature

See also:

• Wikipedia, Maschke’s theorem