Corollary 1

Let G be a finite commutative group. The dual group $\widehat{G}$ is isomorphic to G.

The group G is isomorphic to a finite product of cyclic groups (A, VII, p. 22, th. 3), and its dual group is isomorphic to the product of the dual groups of these (prop. 2 of II, p. 206). One is therefore reduced to the case where G is cyclic, which comes under proposition 17 since the group $\boldsymbol{\mu}_n$ is cyclic of order $n$ (A, V, p. 75, th. 1).