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\title{\textbf{Homework 5}}

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\section*{Section 4.5}
\vskip2ex
\textbf{Exercise 1(a, b, c, e).} \vskip1ex
Prove or disprove each of the following statements.
\begin{enumerate}
\item[(a)]
All of the generators of $\Z_{60}$ are prime.
\item[(b)]
$U(8)$ is cyclic.
\item[(c)]
$\Q$ is cyclic.
\item[(e)]
A group with a finite number of subgroups is finite.
\end{enumerate}
\emph{Solution:}
\hfill $\square$
\vskip2ex


\textbf{Exercise 2(a, b, d, f).}
\vskip1ex
Find the order of each of the following elements.
\begin{enumerate}
\item[(a)]
$5 \in \Z_{12}$
\item[(b)]
$\sqrt{3} \in \R$
\item[(d)]
$-i \in \C^\ast$
\item[(f)]
$312 \in \Z_{471}$.
\end{enumerate}
\emph{Solution: }
\hfill $\square$
\vskip2ex


\textbf{Exercise 3(b, c, h, k).}
\vskip1ex
List all of the elements in each of the following subgroups.
\begin{enumerate}
 \item[(b)]
 The subgroup of $\Z_{24}$ generated by 15
 \item[(c)]
 All subgroups of $\Z_{12}$
 \item[(h)]
The subgroup generated by 5 in $U(18)$
 \item[(k)]
   The subgroup of $\C^\ast$ generated by $2i$
  \end{enumerate}
\emph{Solution: }
\hfill $\square$
\vskip2ex


\textbf{Exercise 4(d).}
\vskip1ex
Find the subgroups of $GL_2(\R)$ generated by the matrix $\begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix}$ \vskip1ex
\emph{Solution: }
\hfill $\square$
\vskip2ex


\textbf{Exercise 11.}
If $a^{24} =e$ in a group $G$, what are the possible orders of $a$?
\vskip1ex
\emph{Solution: }
\hfill $\square$
\vskip2ex


\textbf{Exercise 23(a, c).}
Let $a, b \in G$.  Prove the following statements.
\begin{enumerate}
 \item[(a)]
The order of $a$ is the same as the order of $a^{-1}$. 
 \item[(c)]
The order of $ab$ is the same as the order of $ba$.
\end{enumerate}
\emph{Proof: }
\hfill $\square$
\vskip2ex


\textbf{Exercise 26.}
Prove that  $\Z_p$ has no nontrivial proper subgroups if $p$ is prime.
\vskip1ex
\emph{Proof: }
\hfill $\square$
\vskip2ex


\textbf{Exercise 31.}
Let $G$ be an abelian group. Show that the elements of finite order in $G$ form a subgroup.
This subgroup is called the \textbf{torsion subgroup} of $G$.
\vskip1ex
\emph{Proof: }
\hfill $\square$
\vskip2ex


\textbf{Exercise 32.}
Let $G$ be a finite cyclic group of order $n$ generated by $x$. 
Show that if $y = x^k$ where gcd$(k,n)=1$ then $y$ must be a generator of $G$.
\vskip1ex
\emph{Proof: }
\hfill $\square$
\vskip2ex


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