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\begin{center} {\large \textbf{Math 521 -- Homework 1}} \medskip \\ Due Thursday, August 29, 2019 at 10:15am\ \\ \ \\ 
 \end{center}
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\begin{prob} Let $p\in \Z_+$ be a prime number. 
\begin{itemize}
\item[(a)] Show that the set of non-zero elements of $\Z/p\Z$ form a group under multiplication. 
\item[(b)] Show that this does not hold for $p=6$.
\end{itemize}
\end{prob} 

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\begin{prob} For this exercise, the dihedral group $D_{2n}$ has the usual presentation  \[D_{2n} = \langle r,s | r^n=s^2=1, rs=sr^{-1} \rangle.\] 
\begin{itemize} 
\item[(a)] (DF Ch 1.2 Exercise 2) Use the generators and relations above to show that if $x$ is any element of $D_{2n}$ which is not a power of $r$, then $rx = xr^{-1}$. \medskip 
\item[(b)] (DF Ch 1.2 Exercise 3) Use the generators and relations above to show that every element of $D_{2n}$ which is not a power of $r$ has order $2$.  Deduce that $D_{2n}$
is generated bu the two elements $s$ and $sr$, both of which have order $2$. 
\end{itemize}
\end{prob} 

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\begin{prob}[DF Ch 1.6 Exercise 20]
Let $G$ be a group and let $\Aut(G)$ be the set of all isomorphisms from $G$ onto $G$.  Prove that $\Aut(G)$ is a group under function composition. (This is called the
\emph{automorphism group} of $G$ and the elements of $\Aut(G)$ are called \emph{automorphisms} of $G$). 
\end{prob} 


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