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Maschke's theorem guarantees that every representation of a finite group G over a field whose characteristic does not divide |G| is completely reducible — it decomposes as a direct sum of irreducible subrepresentations. The proof works by averaging an arbitrary inner product over the group to produce a G-invariant one, then taking orthogonal complements. This theorem is the foundation that makes the entire decomposition theory of finite group representations possible.
Maschke's theorem answers the most important structural question in finite group representation theory: can every representation be broken into irreducible pieces? The answer is yes, provided the field's characteristic does not divide the group order. Over ℂ (or ℚ, or ℝ), this condition is always satisfied for finite groups, so every complex representation of a finite group is completely reducible.
The proof uses an averaging trick that is characteristic of the subject. Suppose W ⊆ V is a G-invariant subspace. We want to find a G-invariant complement U with V = W ⊕ U. Start with any linear projection π: V → W (this exists by linear algebra, without any G-equivariance). Now average over the group: define π̃(v) = (1/|G|) Σ_{g∈G} ρ(g) π(ρ(g)⁻¹ v). One checks that π̃ is still a projection onto W, and crucially, π̃ commutes with the G-action. The kernel of π̃ is therefore a G-invariant complement to W. The division by |G| is where the characteristic hypothesis enters — in characteristic p dividing |G|, this division is undefined, and the theorem genuinely fails.
The consequence is that every finite-dimensional representation over ℂ can be written as V ≅ V₁^{⊕n₁} ⊕ V₂^{⊕n₂} ⊕ ··· ⊕ Vₖ^{⊕nₖ}, where V₁, …, Vₖ are pairwise non-isomorphic irreducible representations and the multiplicities n₁, …, nₖ are uniquely determined. This is the representation-theoretic analogue of unique prime factorization. The classification problem for all representations thus reduces to: (1) find all irreducible representations, and (2) determine the multiplicities when a given representation is decomposed. Character theory, built on Schur's lemma and Maschke's theorem, solves both problems.
When the characteristic does divide |G|, we enter modular representation theory, where complete reducibility fails and indecomposable representations need not be irreducible. This is a deeper and more difficult subject, pioneered by Richard Brauer, that requires fundamentally different techniques. Maschke's theorem thus marks the boundary between the "nice" semisimple world and the "wild" modular world.