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\begin{document}

\begin{center}
{\Large Advanced Algebra I}\\
 Homework 1\\
 due on Sep. 29, 2006
\end{center}


\begin{enumerate}
\item Complete the uncompleted proof in the lecture.

\item Let $A$ be an infinite set and for $i=1,2,...$, $|B_i| \le
|A|$. Then $|\amalg_{i=1,2,...} B_i | \le |A|$.

\item Let $A$ be an infinite set and  $\Sigma$ be the set of
finite subsets of $A$. Show that $|Sigma|=|A|$.

\item Let $F$ be an infinite field. Then  $|F|=|F[x]|=|F(x)|$.

\item Let $N,K$ be subgroups of $G$. If $xy=yx$ for all $x \in N,
y \in K$ and $NK=G$, $N \cap K = \{e\}$. Then $G \cong N \times
K$.

\item Let $V$ be a $n$-dimensional vector space over $\bR$. Let
$GL(V)$ denoted the set of invertible linear transformation from
$V$ to $V$. It's clear that $GL(V)$ is a group under the
composition. (One may think it as $GL(n,\bR)$, the groups of
invertible $n \times n$ matrices.) Let $SO(n,\bR):=\{ A  \in
GL(n,\bR)| A A^{t} = I\}$. Show that $SO(n, \bR)$ is a subgroup.
Is it normal?

\item Let $T, D, I$ be the subgroups of $SO(3,\bR)$ that preserves
regular tetrahedron, octahedron, icosahedron respectively.
Determine their cardinality. Can you say something about the
structure of these groups?
\end{enumerate}

\end{document}
\end