Been working on this for a while and haven't gotten anywhere. I would really appreciate some hints.
Let $G$ be a group, not necessarily finite. $V=\mathbb C [G] $ a vector space with basis $(e_g, g\in G)$. Let $U\subseteq V$ have basis $(e_{gh}-e_{hg}, g,h\in G)$.
Show that if dim$V/U=n$, then $G$ has exactly $n$ conjugacy classes.