Maschke's theorem

Maschke's theorem states that every finite-dimensional representation of a finite group over a field whose characteristic does not divide the group order is completely reducible.

Invariant subspace: A subspace preserved by every operator in a group representation. The theorem splits each invariant subspace off as a direct summand.
Group representation: A group action by invertible linear transformations on a vector space. Maschke's theorem concerns finite-dimensional representations of finite groups.
Modular representation theory: The study of representations over fields whose characteristic divides the group order. In this setting, representations need not decompose into irreducibles.
Heinrich Maschke: A German mathematician known for foundational work on representations of finite groups. Maschke proved the complete-reducibility result that bears his name.
Group averaging: A construction that sums or averages an object over a finite group action. Averaging a projection produces one that commutes with the group action.
Irreducible representation: A nonzero representation with no proper nonzero invariant subspaces. Complete reducibility means decomposing a representation into these indivisible pieces.
Failure of Maschke's theorem for cyclic groups: Examples of nonsemisimple representations of cyclic groups when the field characteristic divides the group order. These examples show the hypothesis can fail even for the simplest finite groups.
Ferdinand Georg Frobenius: A German mathematician who helped establish the character theory of finite groups. His character-theoretic framework developed alongside the theorem's early representation theory.
Reynolds operator: An averaging operator that projects onto the invariant part of a group representation. It is the standard averaging construction underlying the theorem's projection argument.
Completely reducible representation: A representation expressible as a direct sum of irreducible representations. This is the exact property guaranteed under the theorem's hypotheses.

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