\documentclass[11pt]{article}
\usepackage{latexsym,amsmath,amssymb}
\usepackage[dvips]{graphicx}
\usepackage[all]{xy}
\usepackage{supertabular}
\usepackage{longtable}
\usepackage{multirow}
\usepackage{epic}
\setlength{\textwidth}{6.5in}
\setlength{\oddsidemargin}{0in}
\setlength{\textheight}{8.0in}
\begin{document}
\centerline{MATH 400 -- Directed Readings in Module and Representation Theory}
\centerline{Final Problem Set}
\centerline{{\bf due:} By email or hardcopy, no later than Saturday, May 15}
\vskip 18pt
\noindent
{\it Groundrules:}  You may consult only our text, Dummit and Foote, {\it Abstract Algebra}.
If you use a result from a chapter we have not discussed, please ``footnote'' it.
\vskip 18pt
\noindent
I.  Let $K,K',P,P',M$ be modules over a ring $R$.  Assume that $P$ and $P'$ are projective
$R$-modules and that we have short exact sequences
$$0 \longrightarrow K \longrightarrow P \longrightarrow M \longrightarrow 0$$
and   
$$0 \longrightarrow K' \longrightarrow P' \longrightarrow M \longrightarrow 0$$
(with the same $M$ in the right-most nonzero position in both).  In this
problem you will show that $K \oplus P' \simeq K' \oplus P$ as $R$-modules.
\begin{enumerate}
\item[A.]  Using the fact that $P$ is projective show that there is a homomorphism
$w : P \to P'$ and
a diagram
\[
\SelectTips {cm}{}
\xymatrix{
0 \ar[r] & K \ar[r]^i  & P \ar[r] \ar[d]^w & M \ar[d]^{id} \ar[r] & 0 \\
0 \ar[r] & K' \ar[r]_j & P' \ar[r]       & M \ar[r] & 0
}
\]
in which the square commutes.
\item[B.]  Continuing from part A, show that there is a homomorphism $u : K \to K'$
such that the left square in 
\[
\SelectTips {cm}{}
\xymatrix{
0 \ar[r] & K \ar[r]^i \ar[d]^u & P \ar[r] \ar[d]^w & M \ar[d]^{id} \ar[r] & 0 \\
0 \ar[r] & K' \ar[r]_j & P' \ar[r]       & M \ar[r] & 0
}
\]
is also commutative.
\item[C.]  Show that the sequence 
$$0 \longrightarrow K \longrightarrow P\oplus K' \longrightarrow P' \longrightarrow 0$$
is exact, where the map $K \longrightarrow P \oplus K'$ is $x \longmapsto (i(x),u(x))$
and the map $P\oplus K' \longrightarrow P'$  is $(y,z) \longmapsto w(y) - j(z)$.
\item[D.]  Deduce that $K \oplus P' \simeq K' \oplus P$ as $R$-modules.
\end{enumerate}
\vskip 10pt
\noindent
II.  Let $F$ be a field and let $R = M_{n\times n}(F)$.  
\begin{enumerate}
\item[A.]  Show that there is a unique irreducible $R$-module $M$, up to isomorphism.
\item[B.]  Let $M$ be the module from part A.  Show that $Hom_R(M,M)$ is isomorphic
to $F$ as a ring.
\end{enumerate} 
\vskip 10pt 
III.  Let $F$ be a field and let $R = F[x]/\langle f(x)\rangle$ for some $f(x) \in F[x]$.  
\begin{enumerate}
\item[A.]  State and prove a necessary and sufficient condition on $f(x)$ for $R$ 
to be a semisimple ring.
\item[B.]  Describe the Wedderburn decomposition of $R$ in the case that $R$
is semisimple.  (That is, what are the $n_i$ and the $\Delta_i$ in the isomorphism
$$R \simeq M_{n_1\times n_1}(\Delta_1) \times \cdots \times M_{n_k \times n_k}(\Delta_k)$$
given by Wedderburn's Theorem?) 
\end{enumerate} 
\vskip 10pt
\noindent
Read section 19.1, and using the ideas presented there do the following problems:
\vskip 10pt
\noindent
IV.  Show that the character table of any finite group is an invertible
$k\times k$ matrix, where $k = $ the number of conjugacy classes, or 
the number of irreducible characters.   
\vskip 10pt
\noindent
V. Compute the character tables of 
\begin{enumerate}
\item[A.]  The group $S_3 \times S_3$.  
\item[B.]  The alternating group $A_5$.
\end{enumerate}  
\end{document}
