A criterion in representation theory stating that for a finite group G and a field k whose characteristic does not divide |G|, the group algebra k[G] is semisimple; equivalently every finite-dimensional k-representation of G decomposes as a direct sum of irreducible representations.

Principle

If the order of the group is invertible in the base field, one can average linear maps over the group to produce G-invariant complements, ensuring complete reducibility of modules over the group algebra.

Demonstration

Over the complex numbers (characteristic 0), any finite group representation decomposes into irreducibles; concretely, the regular representation C[G] splits as a direct sum of matrix algebras corresponding to irreducible characters, because 1/|G| times the sum over G provides projections onto isotypic components.

Misapplication

Assuming Maschke's decomposition in the modular case (when char(k) divides |G|) leads to erroneous conclusions: indecomposable modules need not be irreducible and averaging by 1/|G| is invalid.

Consequence

When hypotheses hold, the representation theory of G reduces to studying simple modules and their matrix algebras; the group algebra has zero Jacobson radical and satisfies Artin–Wedderburn structure, simplifying classification of representations and characters.

Reversal

If char(k) divides |G|, k[G] can have a nonzero radical and modules need not split; modular representation theory then features extensions, projective covers, and indecomposables that are not simple.

Boundary

Applies to group algebras of finite groups and finite-dimensional representations over fields; it does not cover infinite groups, infinite-dimensional representations, or fields whose characteristic divides the group order.

Semantic Tension

Maschke's Theorem is often conflated with general semisimplicity criteria for algebras (e.g., Artin–Wedderburn); the tension lies between an arithmetic invertibility condition (|G| invertible) and more structural algebraic notions of semisimplicity in broader contexts.

Synthesis

Maschke's Theorem ties a simple arithmetic hypothesis (the group order is a unit in the field) to a powerful structural conclusion (semisimplicity of k[G] and full reducibility of representations), achieved concretely by averaging to produce invariant complements.