For a topological group G, we denote by G _a the arc component of the neutral element and by G^Ù{G^\wedge} the character group of G, i.e. the group of all continuous homomorphisms from G into T. We prove the following theorem: Let G be a connected locally compact abelian group and let i: G_a ® G{\iota : G_a \rightarrow G} be the embedding. Then i^Ù : G^Ù ® (G_a)^Ù, c® c°i{\iota^\wedge : G^\wedge \rightarrow (G_a)^\wedge, \chi \mapsto \chi \circ \iota} is a topological isomorphism. In particular, the character group of the arc component of a compact abelian group is discrete. Some conclusions will be drawn.