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\noindent{\Large{\textsc{Modern Algebra \hfill Assignment 12.1  \\ Spring 2019 \hfill Due May 1}}}

\large

\vspace{1cm}


\begin{exercise}
Given $n \in \N$ at least 3, let $\zeta_n = \cos(2 \pi i/n) + i \sin (2 \pi i/n) = e^{\frac{2 \pi i}{n}}$.  Define $R, F \in \operatorname{Perm}(\C)$ by $R(z) = \zeta_n z$ and $F(z) = \overline{z}$.  Prove that the subgroup $\langle R, F \rangle$ of $\operatorname{Perm}(\C)$ is isomorphic to $D_n$.
\end{exercise}

\begin{exercise}  
Find all normal subgroups of $D_n$.
\end{exercise}

\begin{exercise}
Regular polygons are not the only geometric objects with symmetry groups.  Determine the symmetry group of the regular tetrahedron.
\end{exercise}


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