= Solution
<Maschke's theorem> makes the <group algebra> $\mathbb CG$ semisimple, since $G$ is a <finite group> and the ground field has <characteristic zero>. By the <Artin–Wedderburn theorem>, write $\mathbb CG\cong\prod_{i=1}^r M_{d_i}(\mathbb C)$. Its <simple modules> give exactly the $r$ nonisomorphic irreducible complex <group representations>.
The center of each <matrix algebra> consists of scalar matrices, so $\dim_{\mathbb C}Z(\mathbb CG)=r$. On the other hand, an element $\sum_g a_g g$ is central precisely when its coefficients are constant on <conjugacy classes>. The sums of the elements in the separate <conjugacy classes> are therefore a <basis> of the center. Hence \b[the number of irreducible complex representations equals the number of conjugacy classes].
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