Let $G$ be a topological group. Let $C$ be any connected component of $G$. It seems that $C$ may not be a subgroup of $G$ unless $C$ contains the identity $1$ of $G$. However, is it true that $C$ has a structure of a group on its own?
For example, let $A$ be any finitely generated abelian group. Consider the character group $\widehat{A}:=\mathrm{Hom}(A,\Bbb C^\times)$. As groups, $\widehat{A}$ is isomorphic to $F \times (\Bbb C^\times)^{r}$ with $F$ is a finite abelian and $r=\mathrm{rank}(A)$. Is it enough to say that $\widehat{A}$ is a $topological$ group which has a finite number of connected components? It seems that each component is isomorphic (as groups) to a torus $(\Bbb C^\times)^{r}$ . And is it true that each connected component itself is a group?