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\begin{document}
\section*{Lior Silberman's Math 501: Problem Set 1 (due 18/9/2020)}
Practice problems, any sub-parts marked ``OPT'' (optional) and supplementary
problems are not for submission. RMK are remarks. Starred problems
are more difficult.
\begin{center}
\textbf{Review of group theory}
\par\end{center}
\begin{lyxlist}{10.}
\item [{1.}] (Cyclic groups)
\begin{lyxlist}{10.}
\item [{(a)}] Which groups have no non-trivial proper subgroups?
\item [{(b)}] Show that the infinite cyclic group $\Z$ is the unique group
which has non-trivial proper subgroups and is isomorphic to all of
them.
\item [{\bigskip{}
}]~
\end{lyxlist}
\item [{2.}] (Groups with many involutions) Let $G$ be a finite group,
and let $I=\left\{ g\in G\mid g^{2}=e\right\} \setminus\left\{ e\right\} $
be its subset of \emph{involutions} ($e$ is the identity element
of $G$).
\begin{lyxlist}{10.}
\item [{(a)}] Show that $G$ is abelian if it has \emph{exponent $2$},
that is if $G=I\cup\left\{ e\right\} $.
\item [{({*}{*}b)}] Show that $G$ is abelian if $\left|I\right|\geq\frac{3}{4}\left|G\right|$.\bigskip{}
\end{lyxlist}
\item [{3.}] Fix a set $X$. The \emph{support} of a permutation $\sigma\in S_{X}$
is the set $\supp(\sigma)=\left\{ x\in X\mid\sigma(x)\neq x\right\} $.
\begin{lyxlist}{10.}
\item [{(a)}] Let $F_{X}\subset S_{X}$ be the set of permutations of finite
support. Show that $F_{X}$ is a normal subgroup.
\item [{(b)}] Show that $F_{X}$ is generated by transpositions, and that
there is a homomorphism $\sgn\colon F_{X}\to\left\{ \pm1\right\} $
taking the value $-1$ on all transpositions. Write $A_{X}$ for its
kernel.
\item [{({*}c)}] Suppose $X$ is infinite. Show that $A_{X}$ is a simple
group. You may use the fact that $A_{n}$ are simple for $n\geq5$.
\bigskip{}
\end{lyxlist}
\end{lyxlist}
\begin{center}
\textbf{Composition series and solvable groups}
\par\end{center}
\begin{lyxlist}{10.}
\item [{4.}] Find a group which has no composition series.\bigskip{}
\item [{5.}] Show that every group of order $p^{2}q^{2}$ is solvable.\bigskip{}
\item [{6.}] Let $R$ be a ring. Let $G=\GL_{n}(R)$ be the group of invertible
$n\times n$ matrices with entries in $R$, let $B