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Let $G=\{g_1,g_2,\dots,g_n\}$ be an arbitrary finite group. We consider its representations over $\mathbb{C}$. There is Maschke's theorem which states that each representation of $G$ is a direct sum of irreducible ones. I'm trying to link this result with some properties of the group algebra $\mathbb{C}[G]$: it should be the sum of matrix algebras.

So, the regular representation, induced by the action of $G$ on $\{e_{g_1},e_{g_2},\ldots,e_{g_n}\}$ (each $g$ sends $e_h$ to $e_{gh}$) is a representation of $G$ in the group algebra, i.e. the homomorphism $G\rightarrow \mathrm{GL}(\mathbb{C}[G])$. Maschke's theorem allows us to decompose $\mathbb{C}[G]$ in the sum of irreducible representations: $V_1^{\bigoplus m_1}\bigoplus\cdots\bigoplus V_k^{\bigoplus m_k}$ (each $V_i$ is irreducible).

Well, where are matrix algebras...?

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You will need Schur's lemma to prove this. A proof reference can be found here, Lemma 9, Theorem 10.

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