\documentclass[11pt]{amsart}
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\begin{document}
\title{Math 251: Abstract Algebra I \\ Review, Exam \#2}
\date{31 October 2007; exam 7 November 2007}
\maketitle
\begin{problab}{1}
Let $G$ be a group, $H$ a subgroup, and $N$ a normal subgroup. Let $HN=\{hn:h \in H,n \in N\}$. Prove that $HN$ is a subgroup of $G$.
\end{problab}
\begin{problab}{2}
\begin{enumalph}
\item Let $G$ be a group, and suppose that $G$ has normal subgroups of orders $2$ and $5$. Show that $G$ contains an element of order $10$.
\item Give an example to show that a group $G$ may have elements of orders $2$ and $5$ but no element of order $10$.
\end{enumalph}
\end{problab}
\begin{problab}{3}
Let $H=\{\sigma \in S_6:\sigma(4)=4\}$. Show that $H$ is \emph{not} a normal subgroup in $S_6$.
\end{problab}
\begin{problab}{4}
Show that $\Inn(G) = \{1\}$ if and only if $G$ is abelian.
\end{problab}
\begin{problab}{5}
Show that $Z(S_n)=\{()\}$ if $n \geq 3$.
\end{problab}
\begin{problab}{6}
Draw the lattice of subgroups for $\Z/24\Z$.
\end{problab}
\begin{problab}{7}
\begin{enumalph}
\item Let $G$ be an abelian group and let $H$ be a subgroup of $G$. Prove that $G/H$ is abelian.
\item Give an example of a non-abelian group $G$ containing a proper normal subgroup $N$ such that $G/N$ is abelian.
\end{enumalph}
\end{problab}
\begin{problab}{8}
Prove that $S_4$ has no subgroup isomorphic to $Q_8$.
\end{problab}
\begin{problab}{9}
Find all conjugacy classes and their sizes in $A_4$.
\end{problab}
\end{document}