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Maschke's theorem

Maschke’s theorem

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

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

Theorem

If G1 k|G|1_k is invertible in kk, then an exact sequence of k[G]k[G]-modules splits iff it splits after applying the forgetful functor from k[G]k[G]-modules to kk-modules (and the splitting in k[G]Mod{}_{k[G]}Mod can be functorially constructed from the splitting in kMod{}_{k}Mod).

If kk is a field, it follows that the k[G]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?).

Revised on June 22, 2011 23:49:14 by Anonymous Coward (128.97.4.167)