Character Group

This is a special case of the character of a representation where the vector space is one dimensional.

Proof

Suppose $f,f^{\prime }\in \hat{G}$ and let $h=f\times f^{\prime }$, 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 $(f\times f^{\prime })(a)=f(a)f^{\prime }(a)=f^{\prime }(a)$ for all $f^{\prime }\in \hat{G}$ 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