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\title{Math 420-620\\Fall 2012\\Homework 3}
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\centerline{\it Due Monday September 17, 2012.}

\centerline{}




\noindent 1. (5 pt) Consider the map $\phi:G\longrightarrow G$ given by $\phi(x)=x^{-1}$.
\begin{itemize}
\item[a)] (5 pt) Show that $\phi$ is a homomorphism if and only if $G$ is abelian.
\item[b)] (5 pt) Show that if $\phi$ is a homomorphism, then $\phi$ is an automorphism. 
\end{itemize}





\centerline{}

\noindent 2. Consider the group, $G$, generated by the two matrices

\[ \left( \begin{array}{cc}
0 & 1 \\
-1 & 0
\end{array} \right)
,
\left( \begin{array}{cc}
0 & 1 \\
1 & 0
\end{array} \right)
\]

\begin{itemize}
\item[a)] (5 pt) Find the order of $G$.
\item[b)] (5 pt) Is $G$ isomorphic to the quaternion group $\text{Q}_8$? Why or why not?
\end{itemize}


\centerline{}

\noindent 3. Let $G$ be a group. We define $\text{Aut}(G)=\{\phi:G\longrightarrow G\vert\text{ $\phi$ is an automorphism.}\}$.
\begin{itemize}
\item[a)] (5 pt) Show that $\text{Aut}(G)$ is a group.
\item[b)] (5 pt) Suppose we define $\phi_g:G\longrightarrow G$ by $\phi_g(x)=gxg^{-1}$. Show that $\phi_g\in\text{Aut}(G)$.
\item[c)] (5 pt) Consider the collection of all $\phi_g, \ g\in G$ (we call this collection $\text{Inn}(G)$). Show that $\text{Inn}(G)$ is a subgroup of $\text{Aut}(G)$. Is it a normal subgroup of $\text{Aut}(G)$?
\item[d)] (5 pt) Show that the correspondence $g\longrightarrow\phi_g$ is a homomorphism from $G$ to $\text{Aut}(G)$. What is its kernal?
\end{itemize}



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