Maschke's averaging lemma
This fact is related to: linear representation theory
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Let be a finite group and a field whose characteristic does not divide the order of . Let be a representation of over . If is an invariant subspace for , then there exists an invariant subspace of , that is complementary to . In other words, any subrepresentation is a direct summand of the whole representation.
The idea is to take any complementary subspace to in , and consider the induced projection from to . Call this projection .
Now, from the representations of on and , we also get a representation of on . Call this representation . Then consider the sum:
Note that the expression is well-defined iff the order of is not a multiple of the characteristic of .
It turns out that is identity restricted to for every , hence the average is also identity when restricted to . Further, the image of under this map is entirely in . Hence the map is a projection from to .
The kernel of that projection is , and that kernel is easily seen to be invariant.