Remark

Let $n$ be the cardinal of the group G, and let $\mu_n(K)$ be the group of $n$-th roots of unity in K. For every $g\in G$, we have $g^n= 1$; consequently, $\widehat{G}$ can be identified with the group Hom(G$, \mu_n(K)$). The group $\mu_n(K)$ is cyclic of order $n$ (V, §11, No. 2, p. 78, Theorem 1). The group $\widehat{G}$ is therefore isomorphic to the group D(G) = Hom(G$,\mathbf{Q}/\mathbf{Z}$). By VII, §4, No. 9, p. 26, Proposition 10, the group $\widehat{G}$ is isomorphic to the group G, and the mapping that sends an element $g$ of G to the homomorphism $\chi \mapsto \chi (g)$ from $\widehat{G}$ to $K^*$ is an isomorphism from G to $\widehat{\widehat{G}}$.