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\begin{document}
\class{500}
\textbf{Webpage:} \url{http://dunfield.info/500}
\textbf{Office hours:} \input office_hour
\textbf{Textbook:} Dummit and Foote, \emph{Abstract Algebra}, 3rd
edition.
\textbf{Suplemental text}: Charles Rezk, Lecture Notes for Math 500,
posted on our course webpage.
\begin{problems}
\item Let $G$ be the set $\Z^2$, with binary operation defined by
\[
(x_1,y_1)\cdot (x_2,y_2) \defeq (x_1+x_2,\; y_1+y_2+x_1x_2).
\]
Show that $(G,\cdot)$ is a finitely generated abelian group, and
determine its invariant factor form.
\item %(DF 5.4.11)
A subgroup $H\leq G$ is \emph{characteristic} if
$\phi(H)=H$ for every $\phi\in \Aut(G)$.
\begin{enumerate}
\item Prove than every characteristic subgroup is normal. Also, give an
example showing that a normal subgroup need not be characteristic.
\item
Show that if $H,K\leq G$
are characteristic subgroups with $HK=G$ and $H\cap K=\{e\}$, then
$\Aut(G)\approx \Aut(H)\times \Aut(K)$.
\end{enumerate}
\item %(DF 5.4.18)
Recall that a group $H$ is \emph{simple} when its only normal
subgroups are $\{e\}$ and $H$ itself.
Let $G_1,\dots,G_n$ be non-abelian simple groups, and let
$G=G_1\times\cdots \times G_n$. Show that every normal subgroup of
$G$ is of the form $G_I$ for some subset $I\subseteq \{1,\dots,n\}$,
where
\[
G_I=\set{(x_1,\dots,x_n)\in G}{\text{$x_k=e$ for all $k\in I$}}.
\]
% End of Rezk's assignment 4
% Start of Rezk's assignment 5
\item Let $G$ be the set $\Z^3$ with binary operation defined by
\[
(x_1,y_1,z_1)\cdot (x_2,y_2,z_2) \defeq (x_1+x_2,\; y_1+y_2,\; z_1+z_2+y_1x_2).
\]
Show that $G$ is a non-abelian group. Then identify an infinite
cyclic normal subgroup $H\leq G$ such that $G/H$ is isomorphic to a
product of two infinite cyclic groups. (That is, show that $G$ is
an extension of $\Z^2$ by $\Z$.) Is this extension split?
\item % (DF 5.5.1-2)
Suppose $G$ is a semidirect product of $H\unlhd G$ and $K\leq G$,
with $\phi\colon K\ra \Aut(H)$ defined by conjugation. Show that
$C_G(H)\cap K=\ker(\phi)$ and $C_G(K)\cap H=N_G(K)\cap H$.
\item Show that there are exactly 4 distinct
homomorphisms $C_2\ra \Aut(C_{8})$. Prove that the resulting
semidirect products give 4 nonisomorphic groups of order $16$.
\item %(DF 6.1.10)
Show that $D_n$ is nilpotent if and only if $n$ is a power of $2$.
\item Let $G=GL_2(\F_3)$. Determine (i) the subgroups $G^{(k)}$ in
the derived series of $G$, and (ii) the subgroups $Z_k(G)$ in the
upper central series of $G$. In both cases, determine the quotient
groups $G^{(k-1)}/G^{(k)}$ and $Z_k(G)/Z_{k-1}(G)$ up to
isomorphism.
% End of Rezk's assignment 5 and all his assignments about groups.
\end{problems}
Credit: Problems 2, 4, 6, and 8 are from [DF] and Problems 1, 3, 5, and 7 from [R].
\end{document}