Maschke’s theorem

Let G be a finite groupMathworldPlanetmath, and k a field of characteristicPlanetmathPlanetmath not dividing |G|. Then any representation V of G over k is completely reducible.


We need only show that any subrepresentation has a complement, and the result follows by inductionMathworldPlanetmath.

Let V be a representation of G and W a subrepresentation. Let π:VW be an arbitrary projection, and let


This map is obviously G-equivariant, and is the identityPlanetmathPlanetmathPlanetmath on W, and its image is contained in W, since W is invariant under G. Thus it is an equivariant projection to W, and its kernel is a complement to W. ∎

Title Maschke’s theorem
Canonical name MaschkesTheorem
Date of creation 2013-03-22 13:21:16
Last modified on 2013-03-22 13:21:16
Owner bwebste (988)
Last modified by bwebste (988)
Numerical id 9
Author bwebste (988)
Entry type Theorem
Classification msc 20C15