Maschke's theorem

Reformulation and the meaning

One of the approaches to representations of finite groups is through module theory. Representations of a group G are replaced by modules over its group algebra K[G]. Irreducible representations correspond to simple modules. Maschke's theorem addresses the question: is a general (finite-dimensional) representation built from irreducible subrepresentations using the direct sum operation? In the module-theoretic language, is an arbitrary module semisimple? In this context, the theorem can be reformulated as follows:

Maschke's Theorem. Let G be a finite group and K a field whose characteristic does not divide the order of G. Then K[G], the group algebra of G, is semisimple.

The importance of this result stems from the well developed theory of semisimple rings, in particular, the Artin–Wedderburn theorem (sometimes referred to as Wedderburn's Structure Theorem). When K is the field of complex numbers, this shows that the algebra K[G] is a product of several copies of complex matrix algebras, one for each irreducible representation. If the field K has characteristic zero, but is not algebraically closed, for example, K is a field of real or rational numbers, then a somewhat more complicated statement holds: the group algebra K[G] is a product of matrix algebras over division rings over K. The summands correspond to irreducible representations of G over K.

Returning to representation theory, Maschke's theorem and its module-theoretic version allow one to make general conclusions about representations of a finite group G without actually computing them. They reduce the task of classifying all representations to a more manageable task of classifying irreducible representations, since when the theorem applies, any representation is a direct sum of irreducible pieces (constituents). Moreover, it follows from the Jordan–Hölder theorem that, while the decomposition into a direct sum of irreducible subrepresentations may not be unique, the irreducible pieces have well-defined multiplicities. In particular, a representation of a finite group over a field of characteristic zero is determined up to isomorphism by its character.

Proof

Let V be a K[G]-submodule. We will prove that V is a direct summand. Let π be any K-linear projection of K[G] onto V. Consider the map φ : K [ G ] → V given by φ ( x ) = 1 # G ∑ s ∈ G s ⋅ π ( s − 1 ⋅ x ) . Then φ is again a projection: it is clearly K-linear, maps K[G] onto V, and induces the identity on V. Moreover we have

so φ is in fact K[G]-linear. By the splitting lemma, K [ G ] = V ⊕ ker ⁡ φ . This proves that every submodule is a direct summand, that is, K[G] is semisimple.

Converse statement

The above proof depends on the fact that #G is invertible in K. This might lead one to ask if the converse of Maschke's theorem also holds: if the characteristic of K divides the order of G, does it follow that K[G] is not semisimple? The answer is yes.

Proof. For x = ∑ λ g g ∈ K [ G ] define ϵ ( x ) = ∑ λ g . Let I = ker ⁡ ϵ . Then I is a K[G]-submodule. We will prove that for every nontrivial submodule V of K[G], I ∩ V ≠ 0 . Let V be given, and let v = ∑ μ g g be any nonzero element of V. If ϵ ( v ) = 0 , the claim is immediate. Otherwise, let s = ∑ 1 g . Then ϵ ( s ) = # G ⋅ 1 = 0 so s ∈ I and s v = ( ∑ 1 g ) ( ∑ μ g g ) = ∑ ϵ ( v ) g = ϵ ( v ) s so that s v is an element of both I and V. This proves that V is not a direct complement of I for all V, so K[G] is not semisimple.