Non identity character of an Abelian group How does one show that for any non-trivial abelian group, there exists some non-trivial character? 
After looking up for a while, the best solution I could find was Pontrayagin duality, which seems like too heavy a machinery for this problem.

For arbitrary abelian groups something non-trivial has to be used, it seems. For finite abelian groups, though, we have a very elementary Lemma:
Lemma: Let $G$ be a finite abelian group and $a,b$ distinct elements in $G$. Then there exists a character $\chi$ of $G$ such that $\chi(a)\neq \chi(b)$.
For a proof see here, Lemma $1.3.33$. So for $n=|G|\ge 2$ we see that there exists a non-trivial character of $G$.
