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 allow 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.Theorem. If V is a complex representation of a finite group G with a subrepresentation W, then there is another subrepresentation U of V such that V=W⊕U.Corollary (Maschke's theorem). Every representation of a finite group G over a field F with characteristic not dividing the order of G is a direct sum of irreducible representations.Maschke's Theorem. Let G be a finite group and K a field whose characteristic does not divide the order of G. Then K, the group algebra of G, is semisimple.Maschke's theorem. If G is a group and F is a field with characteristic not dividing the order of G, then the category of representations of G over F is semi-simple. 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 allow 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. 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. Maschke's theorem is commonly formulated as a corollary to the following result: