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\begin{document}

\maketitle

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

{\Large\bf HOMEWORK ASSIGNMENT V}\\
\medskip
{\bf (Turn in underlined problems.)}}
\end{center}

\vspace{.5in}

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

\medskip
\noindent
2 \> \underline{2.70}
\end{tabbing}

}
\vspace{.5in}

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

{\bf 

%\underline{1.} Find examples of groups satisfying 
%$N\lhd G$ and $N'\lhd G'$ such that 
%\begin{enumerate}
%\item[(a)] $G\cong G' \& N\not\cong N' \& G/N \cong G'/N'$.
%\item[(b)] $G\not\cong G' \& N\cong N' \& G/N \cong G'/N'$.
%\end{enumerate}

\medskip

\underline{1.}
Let ${\BR}$ be the group of real numbers under addition,
let ${\BZ}$ be the subgroup of integers under addition,
and let $U$ be the multiplicative group of complex
numbers of absolute value one (the circle group).
Use the First Isomorphism Theorem to
show that ${\BR}/{\BZ}\cong U$.

\medskip

\underline{2.}
It is known that  
every group whose order is a prime power
has a nontrivial center. Use this to 
show that every abelian group of size
$p^2$, where $p$ is prime,  is isomorphic to either
${\BZ}_{p^2}$ or ${\BZ}_p\times {\BZ}_p$.

\medskip

\underline{3.} Let $A$ be an abelian group.
Let $T$ be the set of elements of $A$
that have finite order. Show that $T$
is a (normal) subgroup of $A$.
(It is called the {\it torsion subgroup},
as elements of finite order are sometimes
called {\it torsion elements}.)
Show that $A/T$ is {\it torsion free};
i.e., $A/T$ has no elements of finite order.

\medskip

\underline{4.} Show that $S_m\times S_n$ is isomorphic
to a subgroup of $S_{m+n}$.

\medskip

5. Let $\bullet$ denote the one element group. Show
that for any groups $G, H, K$ we have
\begin{enumerate}
\item[(1)] $\bullet \times G\cong G$,
\item[(2)] $G\times H\cong H\times G$, and 
\item[(3)] $G\times (H\times K) \cong (G\times H)\times K$.
\end{enumerate}
Because of these properties we sometimes
think of the isomorphism types of groups as a commutative
monoid under direct product and $\bullet$.
This is a slightly incorrect thing to do.

\medskip

For Graduate Credit:

E. Show that a function $\varphi:G\rightarrow H$ from one
group to another is a homomorphism if and only if 
its graph $\{(g,\varphi(g))\;|\;g\in G\}$ is a subgroup
of $G\times H$. Show that 
the graph of $\varphi$ is a normal subgroup
of $G\times H$ if and only if
$\varphi(G)\subseteq Z(H)$.


\end{document}

