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\title{General Mathematics and Computational Science II}
\author{Exercise 20}
\date{May 7, 2007}
\maketitle

Let $G$ be a finite abelian group of order $n$ (i.e.\ $G$ has $n$
elements), written additively.  Then $\chi \colon G \to \mathbb C
\setminus \{0\}$ is a \emph{character} of $G$ if
\[
  \chi(a+b) = \chi(a) \, \chi(b) 
\]
for all $a,b \in G$.

\begin{enumerate}

\item Show that 
\[
  \chi(-a) = \chi(a)^{-1} = \overline{\chi(a)}
\]
for all $a \in G$.

\emph{Hint:} Recall that the values taken by a character are $n$th
roots of unity.


\item The set of characters of $G$ is called the \emph{dual group},
denoted $\hat G$.  Show that it is indeed an abelian group when the
group operation is given by the product
\[
  (\chi \psi)(a) = \chi(a) \, \psi(a) \,.
\]


\end{enumerate}
\end{document}
