Copies of $\mathbb{C}$ in Artin-Wedderburn decomposition Let $G$ be a finite group, and let  $R = \mathbb{C}G$ be the group algebra. Show that the number of distinct group homomorphisms from $G$ to $\mathbb{C^*}$ equals the number of copies of $\mathbb{C}$ in the Wedderburn-Artin decomposition of $\mathbb{C}G$.
I cannot solve it and my tutor gave me a hint -- $f: G \to \mathbb{C^*}$ extends to a ring homomorphism $f': \mathbb{C}G \to \mathbb{C}$. The $Ker (f')$ is an ideal of $\mathbb{C}G$. But I cannot justify the hint and still have no idea how to proceed. Any help would be greatly appreciated.
 A: Looking at it now it seems simpler than when I last read it.
Maschke's theorem tells us that $\mathbb C[G]$ is a semisimple ring, so every $\mathbb C$ linear homomorphism of $\mathbb C[G]$ into $\mathbb C$ splits it into a copy of $\mathbb C$ complemented by its kernel.
Basic Artin-Wedderburn theorem says that the simple images of $\mathbb C[G]$ necessarily appear in a given Wedderburn decomposition of $\mathbb C[G]$.  So together with the last paragraph, this gives the correspondence of homomorphisms with factors of $\mathbb C$ in the Wedderburn decomposition.
The final thing to convince yourself of is the correspondence between $\mathbb C$-algebra homomorphisms $\mathbb C[G]\to \mathbb C$ and group homomorphisms $G\to\mathbb C^*$.
Of course, any $\mathbb C$ algebra homomorphism $\mathbb C[G]\to \mathbb C$ restricts to a group homomorphism $G\to \mathbb C^*$ because the $g$'s are all units in $\mathbb C[G]$ and have to map to units in $\mathbb C$ (because those nonzero homomorphisms have to map identity to identity.)
The thing that sounds like it is most blocking you is the other direction, that any group homomorphism from $G\to \mathbb C^*$ lifts to one from $\mathbb C[G]\to \mathbb C$.  This is easy to verify, and a standard fact, that $f:G\to \mathbb C^*$ is extended to $\hat f$ by the rule $\hat f(\sum\alpha_gg)=\sum \alpha_gf(g)$.  It is not hard to check that this map from $Hom(G,\mathbb C^*)\to Hom(\mathbb C[G], \mathbb C)$ is inverse to the restriction map, so that you get the correspondence you sought.
