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\begin{center} {\large \textbf{Math 521 -- Homework 2}} \medskip \\ Due Thursday 5, September, 2019 at 10:15am\ \\ \ \\ 
 \end{center}
\bigskip


\begin{prob}[DF Exercises 2.1.6 \& 2.1.7]\  \smallskip 
\begin{itemize}
\item[(a)] Let $G$ be an abelian group. Prove that $\{g\in G \text{ s.t. } |g|<\infty\}$ is a subgroup of $G$ (called the \emph{torsion subgroup} of $G$).  \smallskip 
\item[(b)] Give an explicit example where this set is not a subgroup when $G$ is non-abelian. \smallskip 
\item[(c)] Find the torsion subgroup of $\Z \times \Z/n\Z$. \smallskip 
\item[(d)] Show that the set of elements in $\Z \times \Z/n\Z$ of infinite order together with the identity is \emph{not} a subgroup of this direct product. 
\end{itemize}
\end{prob}

\bigskip  \bigskip 


\begin{prob}
Let $G = {\rm GL}_2(\R)$ and consider the subset of diagonal matrices
\[ A = \left\{ \begin{pmatrix} \lambda_1 & 0 \\ 0 & \lambda_2 \end{pmatrix} \text{ s.t. } \lambda_1, \lambda_2\in \R\backslash\{0\}\right\}.\]
Find $C_G(A)$, $Z(G)$, and $N_G(A)$ (and show that your answers are correct).
\end{prob}

\bigskip \bigskip 

\begin{prob}[DF Exercise 2.4.14]
A group $H$ is called \emph{finitely generated} if there is a finite set $A\subseteq H$ such that $H = \langle A \rangle$. \smallskip 
\begin{itemize}
\item[(a)] Prove that every finite group is finitely generated.\smallskip 
\item[(b)] Prove that $\Z$ is finitely generated. \smallskip 
\item[(c)] Prove that every finitely-generated subgroup of the group $(\Q,+)$ is cyclic. \\
(If $H$ is a finitely-generated subgroup of $\Q$, show that $H\leq \langle 1/k\rangle$, where $k$ is the product of all the denominators
which appear in a set of generators for $H$.)\smallskip 
\item[(d)] Prove that $\Q$ is not finitely generated. 
 \end{itemize}
\end{prob}


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