Maschke's Theorem states that if a finite group is acting on a finite-dimensional vector space over a field whose characteristic does not divide the order of the group, then every representation of the group can be decomposed into a direct sum of irreducible representations. This theorem is fundamental in understanding the structure of representations, as it guarantees that every representation can be analyzed and simplified into simpler components, which is crucial for studying linear representations, matrix representations, and group algebras.
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