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.
 A: This follows from the fact that $GL_1(\mathbb{C})$ is a divisible and hence injective abelian group.  So given any abelian group $A$, a subgroup $B\subseteq A$, and a homomorphism $f:B\to GL_1(\mathbb{C})$, $f$ can be extended to a homomorphism $A\to GL_1(\mathbb{C})$.  Now if $A$ is any nontrivial abelian group, let $B$ be a nontrivial cyclic subgroup of $A$.  It is easy to explicitly construct a nontrivial homomorphism $B\to GL_1(\mathbb{C})$, which then extends to $A$.
(I doubt you will find any simpler proof than this for arbitrary abelian groups--I'm pretty sure this is impossible to prove without using the axiom of choice, for instance.)
A: 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 $G$.
