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\title{Math 332: Homework 2}
%\author{Your name here --- delete the % at the start of the line}
\date{9.II.15}

\begin{document}
\maketitle

\begin{prob}
Show that the following rules constitute (left) group actions on the specified sets:
\begin{enumerate}[(a)]
\item Let $F$ be a field and $F^\times = F\smallsetminus\{0\}$ the multiplicative group of nonzero elements of $F$.  Then $F^\times$ acts on $F$ via $g\cdot a = ga$ where $g\in F^\times$, $a\in F$, and $ga$ is the usual product in $F$ of the two field elements $g$ and $a$.
\item The additive group $\RR$ acts on $\RR^2 = \RR\times \RR$ via $r\cdot (x,y) = (x+ry,y)$.
\item The group $GL_2(\RR)$ of $2\times 2$ invertible matrices with real entries acts on $\RR^2$ via
\[
  \begin{pmatrix} a&b\\ c&d\end{pmatrix}\cdot \begin{pmatrix}x\\ y\end{pmatrix} = \begin{pmatrix} ax+by\\ cx+dy\end{pmatrix}.
\]
(Here we have written $(x,y)\in \RR^2$ as a column vector.)
\end{enumerate}
\emph{Bonus}:  What is the relationship between the action in (b) and the action in (c)?
\end{prob}

\begin{prob}
Prove that the kernel of an action of the group $G$ on a set $A$ is the same as the kernel of the corresponding permutation representation $G\to S_A$.
\end{prob}

\begin{prob}
Assume $n$ is an even positive integer and show that $D_{2n}$ acts on the set consisting of pairs of opposite vertices of a regular $n$-gon.  Find the kernel of this action.
\end{prob}

\begin{prob}
Let $G$ be an abelian group.  Prove that $\{g\in G\mid |g|<\infty\}$ is a subgroup of $G$ (called the \emph{torsion subgroup} of $G$).  Give an explicit example where this set is not a subgroup when $G$ is non-abelian.
\end{prob}

\begin{prob}
Fix some $n\in \ZZ$ with $n>1$.  Find the torsion subgroup of $\ZZ\times (\ZZ/n\ZZ)$.  Show that the set of elements of infinite order together with the identity is \emph{not} a subgroup of this direct product.
\end{prob}

\begin{prob}
Prove that if $H$ and $K$ are subgroups of $G$, then so is their intersection $H\cap K$.  \emph{Bonus}: Prove that the intersection of an arbitrary nonempty collection of subgroups of $G$ is again a subgroup of $G$.  (Do not assume that the collection of subgroups is countable.)
\end{prob}

\begin{prob}
Prove that the subgroup of $GL_2(\FF_3)$ is the subgroup generated by $\begin{pmatrix} 0&-1\\ 1&0\end{pmatrix}$ and $\begin{pmatrix} 1&1\\ 1&-1\end{pmatrix}$ is isomorphic to the quaternion group $Q_8$.
\end{prob}

\begin{prob}
A group $H$ is called \emph{finitely generated} if there is a finite set $A$ such that $H = \langle A\rangle$.

\begin{enumerate}[(a)]
\item Prove that every finite group is finitely generated.
\item Prove that $\ZZ$ is finitely generated.
\item Prove that every finitely generated subgroup of the additive group $\QQ$ is cyclic.  [If $H$ is a finitely generated subgroup of $\QQ$, show that $H\le \langle 1/k\rangle$ where $k$ is the product of all the denominators which appear in a set of generators for $H$.]
\item Prove that $\QQ$ is not finitely generated.
\end{enumerate}
\end{prob}

\begin{prob}
Let $\varphi:G\to H$ be a homomorphism and let $E$ be a subgroup of $H$.  Prove that $\varphi^{-1}(E)\le G$.  If $E\normal H$, prove that $\varphi^{-1}(E)\normal G$.  Deduce that $\ker \varphi \normal G$.
\end{prob}

\begin{prob}
Define $\varphi:\CC^\times\to \RR^\times$ by $\varphi(a+bi) = a^2+b^2$.  Prove that $\varphi$ is a homomorphism and find its image.  Describe the kernel and fibers of $\varphi$ geometrically (as subsets of the plane).
\end{prob}

\begin{prob}
Consider the additive quotient group $\QQ/\ZZ$.
\begin{enumerate}[(a)]
\item Show that every coset of $\ZZ$ in $\QQ$ contains exactly one representative $q\in \QQ$ in the range $0\le q<1$.
\item Show that every element of $\QQ/\ZZ$ has finite order; nevertheless, there are elements of arbitrarily large order.
\item Show that $\QQ/\ZZ$ is the torsion subgroup of $\RR/\ZZ$.
\item Prove that $\QQ/\ZZ$ is isomorphic to the multiplicative group of roots of unity in $\CC^\times$.
\end{enumerate}
\end{prob}

\begin{prob}
Let $SL_n(F) = \{A\in GL_n(F)\mid \det A = 1\}$.  Prove that $SL_n(F)\normal GL_n(F)$ and describe the isomorphism type of the quotient group
\[
   GL_n(F)/SL_n(F).
 \]
 (You may assume that $\det:GL_n(F)\to F^\times$ is a homomorphism.)
\end{prob}

\end{document}