I was reading through Serre's Linear Representation Theory book and encountered a question to show that the set of all irreducible characters of an abelian group form a group.
The proof of closure was given, since if we have two character functions $\chi_1, \chi_2$ of irreducible representations $V,W$ then $\chi_1 * \chi_2$ is the character of the irreducible representation $V \otimes W$.
Additionally the trivial character serves as an identity element.
But I don't see how to take inverses of characters.
If there is a way to define an inverse tensor product, so that given $U,V$ we have that $f(U,V) = Z | Z \otimes V = U$, then I could try to compute the inverse tensor product of a representation, with the trivial representation, and hope that the character laws travel too. But I'm not sure how to define such an operator.