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\maketitle

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

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

\vspace{.5in}

\noindent
{\bf Read Chapter 2.}

\vspace{.5in}

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

\bigskip
\noindent
2 \> \underline{2.4}, \underline{2.6}, 
\underline{2.8}, \underline{2.9}, 2.12, 2.16
\end{tabbing}

}
\vspace{.5in}

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

{\bf 

1. Let $G$ be a group and $H$ a subgroup.
Let $L$ be the equivalent relation on $G$
defined by $x\;L\;y\Longleftrightarrow x^{-1}y\in H$
and let $R$ be the equivalent relation on $G$
defined by $x\;R\;y\Longleftrightarrow xy^{-1}\in H$.
($L$ partitions $G$ into left cosets of $H$ and $R$
partitions $G$ into right cosets of $H$.)
Show that $L$ and $R$ permute. Show that 
$L$-classes and $R$-classes have the same size,
and there are the same number of $L$-classes
as $R$-classes.

\bigskip

\underline{2.} Suppose that $L$ and $R$ are permuting 
equivalence relations on a set $X$.
Suppose that each of the relations 
$L$ and $R$ has exactly $m$
equivalence classes, all of which have size $n$.
(So $|X| = mn$.)
Prove that there is a set 
$\{x_1,\ldots,x_m\}\subseteq X$ which contains exactly
one element from every $L$-class and exactly one element 
from every $R$-class.


\bigskip

3. Prove P. Hall's Theorem that if $H<G$ are finite groups, then
there is a common system of representatives for the left
cosets and the right cosets of $H$ in $G$.


\bigskip


}
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