Character Group
Let \(G\) be an abelian group. A character of \(G\) is a homomorphism from \(G\) to \(\mathbb{C}^\ast\). The set of characters is called the character group denoted by \(\hat{G}\).
This is a special case of the character of a representation where the vector space is one dimensional.
The constant \(1\) function is called the principal character.
The character group is a group under pointwise multiplication of functions.
Proof
Suppose \(f, f' \in \hat{G}\) and let \(h = f \times f'\), the pointwise product. Suppose \(a, b \in G\), then
and thus \(h\) is a homomorphism. Since \(\mathbb{C}^\ast\) is a multiplicative group, we have closure of multiplication and hence \(h\) is a function \(h : G \to \mathbb{C}^\ast\). Thus \(h \in \hat{G}\).
Clearly if \(f\) is the principal character, then \((f \times f')(a) = f(a)f'(a) = f'(a)\) for all \(f' \in \hat{G}\) and for all \(a \in G\), thus we have existence of the identity.
Associativity follows from associativity of multiplication in the group \(\mathbb{C}^\ast\), since multiplication is pointwise.
Similarly if \(f \in \hat{G}\), then \(h = \frac{1}{f}\) is also a function \(G \to \mathbb{C}^\ast\) and