Given a finite group and a field , the group algebra is semisimple iff the characteristic of does not divide order of the group (that is, if is an invertible number in , which is always in characteristic zero).
If the base is just a commutative unital ring, then there is the following statement:
If is invertible in , then an exact sequence of -modules splits iff it splits after applying the forgetful functor from -modules to -modules (and the splitting in can be functorially constructed from the splitting in ).
If is a field, it follows that the 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.