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\begin{center}{\bf M901, Assignment 4: Due Friday, October 7, 2011}\end{center}

{\it Instructions}: Do any three problems.

Background: two categories ${\mathcal C}$ and ${\mathcal D}$ are said to be isomorphic 
if there exist functors $F:{\mathcal C}\to{\mathcal D}$ and 
$G:{\mathcal D}\to{\mathcal C}$ which are mutually inverse to 
each other (i.e., the compositions either way give the identity functors). 
If, however, there are merely natural equivalences between $FG$ 
and the identity functor, and between $GF$ and the identity,
we say that the categories are {\it equivalent}, or that there is an 
{\it equivalence\/} between the categories. 
(Aside: the concept of equivalence of categories turns out to be more useful
than the concept of isomorphism of categories.)

More background: let $X$ be a set and let $R$ be a set of expressions of the form
$x_1^{m_1}\cdots x_r^{m_r}$, where $x_i\in X$ and $m_i\in\{-1,1\}$ for each $i$.
Let $B$ be the free group on the set $X$ (with respect to a map $\phi:X\to B$).
Assume there is a group $G$ and a map $\gamma:X\to G$ such that 
the homomorphism $\Gamma:B\to G$ induced by $\gamma$ is surjective and its
kernel $K$ is the smallest normal subgroup containing 
$\phi(x_1)^{m_1}\cdots \phi(x_r)^{m_r}$ for each expression
$x_1^{m_1}\cdots x_r^{m_r}$ in $R$.
Then we say that $G$ is defined by generators $X$ and relations $R$, and that
$(X, R)$ is a {\it presentation\/} for $G$. If $X$ is finite, we say $G$ is finitely generated.
If $R$ is also finite, we say that $G$ is finitely presented.

One more thing: it is known that two vector spaces (over the same field $K$) are isomorphic
(as vector spaces over $K$) if and only if they have the same dimension
(where the dimension of a vector space is the cardinality of a basis).
For similar reasons, if $B$ is a free abelian group on a set $X$ 
and $C$ is free abelian on a set $Y$, then $B$ and $C$ are isomorphic if and 
only if there is a bijection $X\to Y$. You may assume this if you wish for this problem 
assignment. You may also assume any of the problems below to do any of the others. 


\begin{itemize}

\item[(1)] Let $X$ be a set and define a covariant functor $F_X$ from the category ${\mathcal A}$
of abelian groups to the category ${\mathcal S}$ of sets by
defining $F_X$ on objects by $F_X(C)=Mor_{\mathcal S}(X,C)$
and on morphisms $h:C\to D$ by $(F_X(h))(f)=h\circ f$, which we can write as $h_*(f)$.
If an abelian group $B$ together with a map $\phi:X\to B$ is a free abelian group on the set $X$
(i.e., if it satisfies the universal property that for any map $\Psi:X\to D$
to an abelian group $D$ there is a unique homomorphism $h_\Psi:B\to D$
such that
$$
\begin{matrix}
B &\xrightarrow{h_\Psi} & D  \\
\rlap{$\lower3pt\hbox{$\scriptstyle\phi$}\nwarrow$} & {} & \llap{$\nearrow$\lower3pt\hbox{$\scriptstyle\Psi$}} \\
 & X &
\end{matrix}
$$ 
commutes), prove that $B$ represents the functor $F_X$.
(This is the converse of what we showed in class; we showed that
if $B$ represents $F_X$, then $B$ is a free abelian group on $X$
with respect to an appropriately defined map $X\to B$.)
\vskip\baselineskip

\item[(2)] Let ${\mathcal C}$ be the category whose objects are
the real vector spaces ${\bf R}^n$, for $n=0,1,2,\ldots$ and whose morphisms
are linear transformations between these vector spaces.
Let ${\mathcal V}$ be the usual category of real finite 
dimensional vector spaces. Show that ${\mathcal C}$ and ${\mathcal V}$
are equivalent categories.
\vskip\baselineskip

\item[(3)] Let $X$ be a set with a map $\phi:X\to B$ to a group $B$ such that 
$(B,\phi)$ is a free group on the set $X$. Let $N$ be the subgroup
generated by all commutators of $B$. Recall that $N$ is a normal subgroup of $B$;
thus we have the quotient homomorphism $q:B\to B/N$. Show that $(B/N,q\circ\phi)$
is a free abelian group on $X$.
\vskip\baselineskip

\item[(4)] 
For any set $X$ and any set $R$ of relations, show that there 
is a group $G$ with presentation $(X,R)$. Moreover, if $H$ is another group
with presentation $(X,R)$, show that $G$ and $H$ are isomorphic groups.
\vskip\baselineskip

\item[(5)] Determine the order of the group generated by $X=\{r,f\}$
with relations $R=\{r^3,f^2,rfrf\}$. Justify your answer.
\vskip\baselineskip

\item[(6)] Let $F_i$ be a free group on the set $X_i$ for $i=1,2$, and let
$N_i$ be the commutator subgroup of $F_i$.
\begin{itemize}
\item[(a)] If there is a bijection $X_1\to X_2$ (i.e., if $X_1$ and $X_2$ are isomorphic as sets),
show that $F_1$ and $F_2$ are isomorphic as groups.
\item[(b)] Prove the converse. (If $F_1$ and $F_2$ are isomorphic as groups, show that
so are $F_1/N_1$ and $F_2/N_2$. Now apply (3) to conclude that 
there is a bijection $X_1\to X_2$.)
\end{itemize}
\end{itemize}

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