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\title{Math 251: Abstract Algebra I \\ In Class Review, Exam \#2}
\date{5 November 2007; exam 7 November 2007}
\maketitle
\begin{problab}{A}
How many elements in $S_6$ are conjugate to $(1\ 2\ 3)(4\ 5)$?
\end{problab}
\vspace{3in}
\begin{problab}{B}
Let $N$ be a normal subgroup of a group $G$ of prime index $p$. Show that if $H$ is a subgroup of $G$ with $H \supset N$, then either $H=N$ or $H=G$.
\end{problab}
\newpage
\begin{problab}{C}
Let $G$ be a group and let $g \in G$ be an element of order $2$. Suppose that $G$ acts on a finite set $X$, and consider the resulting permutation representation $\phi:G \to S_X$. Show that $\phi(g)$ can be written as the product of disjoint transpositions.
\end{problab}
\end{document}