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\begin{document}

\begin{center}
{{\Large\bf ALGEBRA 2}\\
\bigskip

{\Large\bf HOMEWORK ASSIGNMENT II}\\

\bigskip

(Turn in underlined problems.)}

\bigskip

Read Section 10.1.
\end{center}

\vspace{.5in}

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

{\bf 

1. Show that universal arrows are unique up to unique
isomorphism. (This means: If $F:{\vr C}\rightarrow {\vr D}$
is a functor, and $(A,m)$ and $(A',m')$ are both
universal arrows from
$X\in {\rm Ob}({\vr D})$ to $F$, then there
is a unique isomorphism $i: A\rightarrow A'$
such that $m'=F(i)\circ m$.)

\bigskip

\underline{2.}
Show that every set is free in the
variety of sets.

\bigskip

3. Show that every vector space over a division ring
$\m D$ is free in the variety of all 
$\m d$-vector spaces.

\bigskip

\underline{4.}
Is the Klein group free in the variety 
that it generates?

\bigskip

5. Show that the free lattice on two generators has size four.
Show that the free lattice on three generators is infinite.
(Hint for the second part:
it is enough to show that there is no finite
bound on the size of $3$-generated lattices.)

\bigskip

6.
\begin{enumerate}
\item[\underline{(a)}]
Find a presentation of $S_3$ relative to the variety
of all groups.
\item[(b)]
Find a presentation of $S_4$ relative to the variety
of all groups.
\item[(c)]
Find a presentation of the quaternion group
relative to the variety of all groups.
\end{enumerate}

\bigskip

\underline{7.}
\begin{enumerate}
\item[(a)] Show that the coproduct 
${\BZ}_2\coprod {\BZ}_2$ in the variety
of groups of exponent four is isomorphic to 
the dihedral group $D_4$. 
\item[(b)] Show that the coproduct 
${\BZ}_2\coprod {\BZ}_2$ in the variety
of all groups is infinite.
\end{enumerate}


}
\end{document}

