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\title{\textbf{Homework 9 Solutions}}

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\section*{Section 10.4}
\textbf{Exercise 1.}\\
Determine whether $H$ is a normal subgroup of a group $G$. If so, write out a Cayley table for the factor group $G / H$.
\vskip1ex
\emph{Solution: }
\begin{itemize}
 \item[(a)] $G = S_4$ and $H = A_4$.
 \item[(b)] $G = A_5$ and $H = \{(1),(123),(132)\}$.
 \item[(c)] $G = S_4$ and $H=D_4$.
 \item[(e)] $G = \Z$ and $H = 5\Z$. 
\end{itemize}
      
\vskip2ex
\textbf{Exercise 5.}
\vskip1ex
Show that the intersection of two normal subgroups is a normal subgroup. 
\vskip1ex
\emph{Proof:}
\vskip1ex
\hfill $\square$

\vskip2ex
\textbf{Exercise 6.}
\vskip1ex
If $G$ is an abelian group, prove that the factor group $G / N$ must also be abelian.
\vskip1ex
\emph{Proof: }
\vskip1ex
\hfill $\square$

\vskip2ex
\textbf{Exercise 8.}
\vskip1ex
If $G$ is cyclic, prove that the factor group $G / H$ must also be cyclic.
\vskip1ex
\emph{Proof: }
\vskip1ex
\hfill $\square$

\vskip2ex
\textbf{Exercise 9.}
\vskip1ex
Prove or disprove: if $H$ and $G / H$ are cyclic, then $G$ is cyclic. 
\vskip1ex
\emph{Proof: }
\vskip1ex
\hfill $\square$

\vskip2ex
\textbf{Exercise 11.}
\vskip1ex
If a group $G$ has exactly one subgroup $H$ of order $k$, prove that $H$ is normal in $G$.
\vskip1ex
\emph{Proof: }
\vskip1ex
\hfill $\square$

%\vskip2ex
%\textbf{Exercise 12.}
%\vskip1ex
%The centralizer of an element $g$ of a group $G$ is the set $C(g) = \{x\in G: xg=gx \}$.
%Show that $C(g)$ is a subgroup of $G$. If $g$ generates a normal subgroup of $G$, prove that $C(g)$ is normal in $G$. 
%\vskip1ex
%\emph{Proof: }
%\vskip1ex
%\hfill $\square$

\vskip2ex
\textbf{Exercise 13.}
\vskip1ex
The center of a group $G$ is the set $Z(G) = \{x\in G: xg=gx \  \forall g \in G\}$.
\begin{enumerate}
 \item Calculate the center of $S_3$.
 \item Calculate the center of $GL_2(\R)$.
 \item Show that the center of any group $G$ is a normal subgroup of $G$.
 \item If $G / Z(G)$ is cyclic, show that $G$ is abelian.
\end{enumerate}
\vskip1ex
\emph{Proof: }
\vskip1ex
\hfill $\square$   

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