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\begin{center}
{{\Large\bf ALGEBRA I}\\
\bigskip

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

\vspace{.5in}

\noindent
{\bf Read Chapter 1.}

\vspace{.5in}

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

\bigskip
\noindent
1 \> 1.12, \underline{1.17}, 1.38, \underline{1.41}, \underline{1.43}, 
\underline{1.47}
\end{tabbing}

}
\vspace{.5in}

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

{\bf 

1. Find an isomorphism from the group ${\cal P}(X\times Y)$
to the group ${\cal P}(X)^Y$. 

\bigskip

\underline{2.} Construct a group with more than one
one element whose elements are binary relations, whose
multiplication is relational composition, and whose
inverse operation is relational converse.
Is it possible that all elements of your group
are equivalence relations?


\bigskip

3. Show that a groups satisfies the law $(xy)^2=x^2y^2$ if
and only if it is abelian. Find a nonabelian group that 
satisfies $(xy)^3=x^3y^3$.


\bigskip

4. Let $\alpha$ equal the $r$-cycle $(1\;2\;\cdots\;r)$ 
in $S_n$. Find all $\beta\in S_n$ for which $\alpha\beta=\beta\alpha$.
(Note: $\beta$ does not have to be a cycle.)



}
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