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Maschke's theorem states that if $G$ is a finite group and $V$ is a finite-dimensional representation of $G$, then $V$ can be decomposed into a direct sum of irreducible representations. The video expresses this as $V = U \oplus W_1 \oplus W_2 \oplus \cdots \oplus W_k$, where each $W_i$ is an irreducible representation, meaning it has no nontrivial $G$-invariant subspace.

Conditions: $G$ is a finite group.; $V$ is a finite-dimensional representation space.; $\rho: G \to GL(V)$ is a representation.

Maschke's theorem states that if $G$ is a finite group and $V$ is a finite-dimensional representation of $G$, then $V$ can be decomposed into a direct sum of irreducible representations. The video expresses this as $V = U \oplus W_1 \oplus W_2 \oplus \cdots \oplus W_k$, where each $W_i$ is an irreducible representation, meaning it has no nontrivial $G$-invariant subspace.

Conditions: $G$ is a finite group.; $V$ is a finite-dimensional representation space.; $\rho: G \to GL(V)$ is a representation.