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\noindent{\Large{\textsc{Modern Algebra \hfill Assignment 2.2  \\ Spring 2023 \hfill Due January 25}}}

\large

\vspace{1cm}

\begin{exercise}
Construct Cayley tables for the dihedral groups $D_3$ and $D_4$.
\end{exercise}

\begin{exercise}
Given $n \ge 3$, let $P \subset \R^2$ be a regular $n$-gon.  Give each vertex of $P$ a unique label (in any way you like).  Let $r : P \to P$ denote the smallest possible counterclockwise rotation of $P$ about its center, and let $f :P \to P$ denote any symmetry of $P$ that reverses the ordering of the labels of the vertices of $P$.

\begin{itemize}

\item[\bf a.]  Explain why $D_n = \{\operatorname{id}, r, r^2, \ldots, r^{n-1}, f, rf, r^2 f, \ldots, r^{n-1} f  \}$.

\item[\bf b.]  Show that $fr = r^{n-1}f$.

\item[\bf c.]  Show that $D_n$ is nonabelian by proving that $rf \ne fr$.  

\end{itemize}

\end{exercise}

\begin{exercise}
Describe the group of symmetries of the doubly infinite string 
\[
\cdots \text{H} \, \text{H} \, \text{H} \, \text{H} \cdots
\]
Is this group abelian?
\end{exercise}

\begin{exercise}
How large is the group of symmetries of a regular tetrahedron?  Is it abelian?
\end{exercise}

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