Show that $R(G)\cong K_0(F[G])$ Let $F$ be a field of characteristic $0$, $G$ a finite group and let $R(G)$ be the additive group of functions $G\to F$ generated by characters of $G$ of degree $1$. 

Question: How can we show that $R(G)\cong K_0(F[G])$ where $K_0(F[G])$ is the Grothendieck group of finitely generated projective $F[G]$-modules? 

Many thanks in advance. 

If $G$ is a finite group and $F$ is a field of characteristic zero, Then the group algebra $F[G]$ is semisimple by maschke's theorem (this is a fact that is demonstrated in any book of representation theory, i.e Fulton Representation Theory) with one matrix ring factor for each conjugacy class of $G$, so $K_0(F[G])\cong \Bbb Z^n$ by the Artin-Wedderburn theorem and morita invariant of $K_0$   and $n$ is the number of conjugacy class of $G$. 
in other the set of finite-dimensional representations $Rep_F(G)$ is monoid under the direct sum, and $Rep_F(G)\cong \Bbb N^n$. Therefore $F(G)=K_0(Rep_F[G])\cong \Bbb Z^n$. (I apologize in advance if my English is not very good, I've done my best attempt)
