Let $G$ be a compact group. Let $C(G)$ denote the set of all continuous functions $G\to \mathbb{C}$ and let $\mu$ denote the normed Haar measure on $G$. Convolution on $G$, $*:C(G)\times C(G)\to C(G)$, is defined by $(f*g)(a)=\int_G f(ab^{-1})g(b)d\mu (b),\quad f,g,\in C(G).$
I know that $C(G)$ with convolution is a Banach algebra.
Allegedly, this algebra is unital iff G is finite. How does one prove this?