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\begin{document}

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\noindent{\Large{\textsc{Modern Algebra \hfill Assignment 4.1  \\ Spring 2019 \hfill Due February 13}}}

\large

\vspace{1cm}

\begin{exercise}
Prove that $Z(\operatorname{GL}_2(\R)) = \{aI \, | \, a \in \R^{\times} \}$.  [{\em Suggestion:}  It is clear that every matrix of the form $aI$ belongs to the center.  To prove the opposite inclusion, commute an element of the center with the matrices $\smat{}{1}{1}{}$ and $\smat{1}{1}{}{1}$.]
\end{exercise}

\begin{exercise}
A group $G$ is called {\em finitely generated} if there exist $x_1, x_2, \ldots, x_n \in G$ so that $G = \langle x_1, x_2, \ldots, x_n \rangle$.  

\begin{itemize}

\item[\bf a.]  Prove that $\Z^n = \underbrace{\Z \oplus \Z \oplus \cdots \oplus \Z}_{n \text{ times}}$ can be generated by $n$ elements, and no fewer (this requires a little bit of linear algebra).

\item[\bf b.]  Prove that $\Q$ is {\em not} finitely generated.  [{\em Suggestion:}  Argue by contradiction.]

\end{itemize}

\end{exercise}

%\begin{exercise}
%Let $A$ be an abelian group, written additively.

%\begin{itemize}

%\item[\bf a.]  Let $a,b \in A$ and $n \in \Z$.  Prove that $n(a+b) = na + nb$.  [{\em Suggestion:}  Treat the cases $n>0$, $n=0$ and $n<0$ separately.  Induction may be helpful.]

%\item[\bf b.]  For $n \in \N$, let  $A[n] = \{ a \in A \, | \, na = 0 \}$.  Prove that $A[n] < A$.  $A[n]$ is called the {\em $n$-torsion subgroup} of $A$.

%\item[\bf c.] Let
%\[
%A_{\text{tors}} = \bigcup_{n \in \N} A[n].
%\]
%Prove that $A_{\text{tors}} < A$.  $A_{\text{tors}}$ is called the {\em torsion subgroup} of $A$.

%\end{itemize}

%\end{exercise}

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