A criterion for complete reducibility of representations of a finite group G over a field k: the group algebra kG is semisimple (equivalently every finite-dimensional representation is completely reducible) precisely when the characteristic of k does not divide the order of G.

Principle

Averaging over the group using a normalized sum produces G-invariant complements when denominators are available; the existence of the averaging idempotent is the organizing idea behind semisimplicity.

Demonstration

If char(k) does not divide |G|, then for any G-subrepresentation U ⊆ V one can average a projection to U over G to produce a G-equivariant projection, splitting the inclusion and yielding V ≅ U ⊕ U'. Thus every representation splits as a direct sum of irreducibles.

Misapplication

Using Maschke when the field characteristic divides |G| (the modular case): averaging fails because |G| is zero in k, and representations need not be completely reducible.

Consequence

Gives semisimplicity of group algebras in the non-modular case, enabling decomposition into simple modules, character theory, and numerous structural results in finite group representation theory.

Reversal

In the modular case (characteristic dividing |G|) one obtains richer structures with indecomposable non-simple modules, projective covers and block decomposition rather than complete reducibility.

Boundary

Applies to finite groups and finite-dimensional representations over fields; the key hypothesis is that char(k) ∤ |G|; infinite groups or infinite-dimensional representations require different tools.

Semantic Tension

Tension between Maschke's semisimplicity (non-modular) and modular representation theory: Maschke gives a clean decomposition when denominators exist, whereas the modular situation produces subtle extensions and blocks.

Synthesis

Maschke's Theorem says that for finite groups over fields whose characteristic does not divide the group order, averaging yields invariant complements and hence complete reducibility of representations; failure of the denominator condition leads to modular complexity.