isomorphism $ \Bbb CG $ modules

Question

Let $ \Bbb CG$ algebra, and let $ x= \sum _{g \in G} g $ . We define the module $ \Bbb Cx=\{cx: c \in C\}$. I must prove that $$ \Bbb C \cong _{ \Bbb CG} \Bbb Cx $$ Any ideas about that..thanks in advance..

Answer 1

Find an isomorphism! The map $\Bbb C\to \Bbb C x$ is determined by where $1$ is sent. You can't really go wrong here: try anything nonzero and it'll work. Then just prove it's a $\Bbb C G$-module isomorphism.