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\title{Math 251: Abstract Algebra I \\ Homework \#4}
\date{19 September 2007; due Wednesday, 26 September 2007}
\maketitle
\section*{Problems (for all)}
\begin{problab}{1 \textsf{(DF 1.4.10)}}
Let
\[ G=\left\{\begin{pmatrix} a & b \\ 0 & c \end{pmatrix}: a,b,c \in \R, ac \neq 0\right\}. \]
Show that $G$ is a group under matrix multiplication.
\end{problab}
\begin{problab}{2}
In this exercise, we show that each of the following groups are mutually non-isomorphic:
\[ \Z/8\Z, \quad (\Z/16\Z)^\times, \quad D_8, \quad Q_8. \]
\begin{enumalph}
\item Show that each of these groups has order $8$.
\item Show that if $G,H$ are groups and $G \cong H$, then $G$ is abelian if and only if $H$ is abelian. Conclude that neither of the first two groups is isomorphic to either of the last two groups.
\item Show that $\Z/8\Z$ has an element of order $8$ but $(\Z/16\Z)^\times$ has no element of order $8$. Conclude that $\Z/8\Z \not\cong (\Z/16\Z)^\times$.
\item Show that every element in $Q_8 \setminus \{1,-1\}$ has order $4$. Conclude that $D_8 \not\cong Q_8$.
\end{enumalph}
\end{problab}
\begin{problab}{3}
Let $m,n \in \Z$ with $n>1$. Consider the map:
\begin{align*}
\phi:\Z/n\Z &\to \Z/n\Z \\
a &\mapsto ma.
\end{align*}
\begin{enumalph}
\item Show that $\phi$ is a homomorphism.
\item Show that if $\gcd(m,n)=1$, then $m$ is an isomorphism.
\end{enumalph}
\end{problab}
\begin{problab}{4}
Let $G$ be a group. Show that the map $\phi:G \to G$ defined by $\phi(a)=a^2$ is a homomorphism if and only if $G$ is abelian.
\end{problab}
\section*{Extra problems (for grad students)}
\begin{problab}{5 \textsf{(DF 1.4.7)}}
Let $p$ be prime. Show that $\#GL_2(\F_p)=p(p-1)(p^2-1)$. [Hint: Subtract the number of $2 \times 2$-matrices which are \emph{not} invertible from the total number of $2 \times 2$ matrices over $\F_p$. You may use the fact that a $2 \times 2$-matrix is not invertible if and only if one row (or column) is a multiple of the other.]
\end{problab}
\end{document}