# 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?).

Last revised on June 22, 2011 at 23:49:14. See the history of this page for a list of all contributions to it.