# 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. If (V, ρ) is a finite-dimensional representation of a finite group G over a field of characteristic zero, and U is an invariant subspace of V, then the theorem claims that U admits an invariant direct complement W; in other words, the representation (V, ρ) is completely reducible. More generally, the theorem holds for fields of positive characteristic p, such as the finite fields, if the prime p doesn't divide the order of G.

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Maschke's Theorem - Reformulation and The Meaning
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... Wiesner Austria 1934 Ilmenau Walter Feist Walter Kluge Germany Rudolf Hermann Rudolf Maschke Czechoslovakia Josef Heller Albert Krauss Czechoslovakia 1935 Krynica Walter Feist Walter Kluge ...
Representation Theory - Branches and Topics - Finite Groups
... This is a consequence of Maschke's theorem, which states that any subrepresentation V of a G-representation W has a G-invariant complement ... Maschke's theorem holds more generally for fields of positive characteristic p, such as the finite fields, as long as the prime p is coprime to the ... Unitary representations are automatically semisimple, since Maschke's result can be proven by taking the orthogonal complement of a subrepresentation ...

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To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.
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