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\begin{center}
{             Assignment 4\\
       Due in class: Fri, Oct 13.
}
\end{center}


\begin{enumerate}
\item
Let $G$ be a finitely generated Abelian group presented by the
following matrix:
\[
A = 
\left(
\begin{array}{cccc}
1 & 2 &3 &4 \\
5 & 6 & 7 &8 \\
4 & 3 &2 &1 
\end{array}
\right) .
\]
\begin{enumerate}
\item
Compute the Smith normal form of $A$.
\item
Write down a description of $G$ as a direct sum of cyclic groups
using the normal form computed above.
\item
What are the Betti numbers and the torsion coefficients of $G$ ?
\end{enumerate}

The purpose of the following two exercises is to make sure that
you have firmly digested the basic notions of categories discussed
in class. Thus, it is important that you write the proofs in full
details, clearly stating what exactly you are proving.

\item
Recall the definitions of fibre product and fibre co-product.
\begin{enumerate}
\item
Show that fibre products exist in the category of Abelian groups.
If $f: X \rightarrow Z, g: Y\rightarrow Z$ are Abelian group
homomorphisms, show that $X \times_Z Y$ is the set of all pairs
$(x,y), x\in X, y \in Y$ such that $f(x) = g(y)$. 
\item
Prove that the pull back of a surjective homomorphism is surjective.
\end{enumerate}


\item
The purpose of this problem is to show
that fibre coproducts exist in the category of Abelian
groups. 
\begin{enumerate}

\item
In this case the fibre coproduct of two homomorphisms,
$f: Z \rightarrow X, g: Z\rightarrow Y$ is the factor group,
$X \oplus_Z Y = (X \oplus Y)/W$, where $W$ is the subgroup consisting
of all elements  $(f(z), -g(z)), z \in Z$.

\item
Prove that the push out of an injective homomorphism is injective.

\end{enumerate}

\end{enumerate}
\end{document}
