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\begin{center}
{{\Large\bf MODERN ALGEBRA II}\\
\bigskip

{\Large\bf HOMEWORK ASSIGNMENT IV}\\
\bigskip
{\bf (Turn in underlined problems.)}}
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\vspace{.5in}

\noindent
{\bf 
\begin{tabbing}
\underline{\large\bf CHAPTER}\makebox[2in]{}\=\underline{\large\bf PROBLEMS}\\

\bigskip
\noindent
2 \> \underline{2.64}, 2.65
\end{tabbing}

}
\vspace{.5in}

\noindent
\underline{\large\bf ADDITIONAL PROBLEMS}
\vspace{.25in}

{\bf 

1. 
Show that the Klein group $V$ is normal
in $S_4$. Show that $S_4/V\cong S_3$.

\bigskip

\underline{2.}
Let $G$ be a group. Show that the ``diagonal''
$\Delta = \{(g,g)\in G\times G\;|\;g\in G\}$
is a subgroup of $G\times G$.
Show that the diagonal subgroup is normal 
if and only if $G$ is abelian.

\bigskip

\underline{3.}
Suppose that $G$ is an abelian group
and that $\Delta$ is the diagonal subgroup
of $G\times G$. Show that $\Delta\cong G$ and 
$(G\times G)/\Delta\cong G$.

\bigskip

\underline{4.}
Show that $SL(2,{\BR})$ is normal in
$GL(2,{\BR})$. Show that 
\[GL(2,{\BR})/SL(2,{\BR})
\cong\langle {\BR};\cdot,{}^{-1},1\rangle.\]

\underline{5.} Find groups $G$ and $G'$ with normal subgroups 
$N\lhd G$, $N'\lhd G'$ such that $G\cong G', 
N\cong N'$, BUT $G/N\not\cong G'/N'$.
(Hint: There are examples among small abelian groups.)

\bigskip

For Graduate Credit:

D. Show that if a group $G$ has three 
normal subgroups 
$H, J$ and $K$ ordered as shown, then $G$ is abelian.
(First show that if $h\in H$ and $j\in J$,
then $hjh^{-1}j^{-1} \in H\cap J$. Conclude that
elements of $H$ commute
with elements of $J$. Apply this 
argument to all pairs
$(H,J), (H,K)$ and $(J,K)$ and finish the proof.)

\bigskip

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