Maschke's Theorem states that if a finite group is represented on a vector space over a field whose characteristic does not divide the order of the group, then every representation of the group decomposes as a direct sum of irreducible representations. Named for Heinrich Maschke, it guarantees that representation theory over such fields reduces to the study of irreducible pieces.
Facts
StatementMaschke's theorem states that if the order of a finite group G is not divisible by the characteristic of a field K, then every finite-dimensional representation of G over K is completely reducible, decomposing as a direct sum of irreducible representations. 1 Classification
Statement FormCharacterization Theorem 1 Connections
Has Statement Form
In Branch
Proved By
Source Maschke's Theorem (Wikipedia)
Sources
1. Maschke's Theorem (Wikipedia)
Wikimedia FoundationLead paragraph, first sentence
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.
Proved By: Heinrich Maschke, Lead section
Maschke's theorem, named after Heinrich Maschke
In Group: Lie Algebras and Representation Theory, Lead section
named after Heinrich Maschke, is a theorem in group representation theory that concerns the decomposition of representations
View the Source 2. Heinrich Maschke, Biography (MacTutor History of Mathematics)
MacTutor History of Mathematics Archive, University of St AndrewsPage textQuote, Page text
He is best known today for Maschke's theorem, which he published in 1899 , which states that if the order of the finite group G G G is not divisible by the characteristic of the field K K K , then the ( finite-dimensional ) K K K -representations of G G G are completely reducible.
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