Show that the 1-dimensional (complex) characters of a finite group $G$ form a group under pointwise multiplication. Denote this group by $\widehat{G}$. Show that if $g \in G$, the map $\chi \mapsto \chi(g)$ from $\widehat{G}$to $\mathbb{C}$ is a character of $\widehat{G}$, hence an element of $\widehat{G}$. What is the kernel of the $\operatorname{map} G \rightarrow \widehat{\widehat{G}}$?

Show that if $G$ is abelian the map $G \rightarrow \widehat{\widehat{G}}$is an isomorphism. Deduce, from the structure theorem for finite abelian groups, that the groups $G$ and $\widehat{G}$are isomorphic as abstract groups.