Maschke's theorem
In mathematics, Maschke's theorem, named after Heinrich Maschke, is a theorem in group representation theory that concerns the decomposition of representations of a finite group into irreducible pieces. Maschke's theorem allows one to make general conclusions about representations of a finite group G without actually computing them. It reduces 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. Formulations 

Maschke's theorem addresses the question: when is a general (finite-dimensional) representation built from irreducible subrepresentations using the direct sum operation? This question (and its answer) are formulated differently for different perspectives on group representation theory. Group-theoretic 

Maschke's theorem is commonly formulated as a corollary to the following result:

Then the corollary is

The vector space of complex-valued class functions of a group  has a natural -invariant inner product structure, described in the article Schur orthogonality relations. Maschke's theorem was originally proved for the case of representations over  by constructing  as the orthogonal complement of  under this inner product. Module-theoretic 
One of the approaches to representations of finite groups is through module theory. Representations of a group  are replaced by modules over its group algebra  (to be precise, there is an isomorphism of categories between  and , the category of representations of ). Irreducible representations correspond to simple modules. In the module-theoretic language, Maschke's theorem asks: is an arbitrary module semisimple? In this context, the theorem can be reformulated as follows:

The importance of this result stems from the well developed theory of semisimple rings, in particular, their classification as given by the Wedderburn–Artin theorem. When  is the field of complex numbers, this shows that the algebra  is a product of several copies of complex matrix algebras, one for each irreducible representation. If the field  has characteristic zero, but is not algebraically closed, for example if  is the field of real or rational numbers, then a somewhat more complicated statement holds: the group algebra  is a product of matrix algebras over division rings over . The summands correspond to irreducible representations of  over . Category-theoretic 

Reformulated in the language of semi-simple categories, Maschke's theorem states

Proofs

Group-theoretic 
Let U be a subspace of V complement of W. Let  be the projection function, i.e.,  for any . Define , where  is an abbreviation of , with  being the  representation of G on W and V.  Then,  is preserved by G under representation : for any , 

so  implies that . So the restriction of  on  is also a representation. By the definition of , for any , , so , and for any , . Thus, , and . Therefore, . Module-theoretic 
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 

Then φ is again a projection: it is clearly K-linear, maps K[G] to V, and induces the identity on V (therefore, maps K[G] onto V). Moreover we have

so φ is in fact K[G]-linear. By the splitting lemma, . 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  define . Let . Then I is a K[G]-submodule. We will prove that for every nontrivial submodule V of K[G], . Let V be given, and let  be any nonzero element of V. If , the claim is immediate. Otherwise, let . Then  so  and 
 

so that  is a nonzero element of both I and V. This proves V is not a direct complement of I for all V, so K[G] is not semisimple. Non-examples 
The theorem can not apply to the case where G is infinite, or when the field K has characteristics dividing #G. For example, 

 Consider the infinite group  and the representation  defined by . Let , a 1-dimensional subspace of  spanned by . Then the restriction of  on W is a trivial subrepresentation of . However, there's no U such that both W, U are subrepresentations of  and : any such U needs to be 1-dimensional, but any 1-dimensional subspace preserved by  has to be spanned by an eigenvector for , and the only eigenvector for that is . Consider a prime p, and the group , field , and the representation  defined by . Simple calculations show that there is only one eigenvector for  here, so by the same argument, the 1-dimensional subrepresentation of  is unique, and  cannot be decomposed into the direct sum of two 1-dimensional subrepresentations. Notes

References

Representation theory of finite groups
Theorems in group theory
Theorems in representation theory