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 formula_1 has a natural formula_1-invariant inner product structure, described in the article Schur orthogonality relations. Maschke's theorem was originally proved for the case of representations over formula_3 by constructing formula_4 as the orthogonal complement of formula_5 under this inner product. Module-theoretic. One of the approaches to representations of finite groups is through module theory. "Representations" of a group formula_1 are replaced by "modules" over its group algebra formula_7 (to be precise, there is an isomorphism of categories between formula_8 and formula_9, the category of representations of formula_1). 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, the Artin–Wedderburn theorem (sometimes referred to as Wedderburn's Structure Theorem). When formula_11 is the field of complex numbers, this shows that the algebra formula_7 is a product of several copies of complex matrix algebras, one for each irreducible representation. If the field formula_11 has characteristic zero, but is not algebraically closed, for example, formula_11 is a field of  real or rational numbers, then a somewhat more complicated statement holds: the group algebra formula_7 is a product of matrix algebras over division rings over formula_11. The summands correspond to irreducible representations of formula_1 over formula_11. 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 formula_19 be the projection function, i.e., formula_20 for any formula_21. Define formula_22, where formula_23 is an abbreviation of formula_24, with formula_25 being the representation of "G" on "W and" "V". Then, formula_26 is preserved by "G  under representation formula_27: for any formula_28, formula_29 so formula_30 implies that formula_31. So the restriction of formula_27 on formula_26 is also a representation. By the definition of formula_34, for any formula_35, formula_36, so formula_37, and for any formula_38, formula_39. Thus, formula_40, and formula_41. Therefore, formula_42. 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 formula_43 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 formula_44 so φ" is in fact "K"["G"]-linear. By the splitting lemma, formula_45. 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 formula_46 define formula_47. Let formula_48. Then "I" is a "K"["G"]-submodule. We will prove that for every nontrivial  submodule "V" of "K"["G"], formula_49. Let "V" be given, and let formula_50 be any nonzero element of "V". If formula_51, the claim is immediate. Otherwise, let formula_52. Then formula_53 so formula_54 and formula_55 so that formula_56 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,