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\pagestyle{fancy} \lhead{{\sf Homework 5 $|$ Due Wednesday, September 25
}} \rhead{\thepage} \cfoot{{\sf Math 4120/6120 $|$ Modern Algebra $|$
    Fall 2019 $|$ M.~Macauley}}



\begin{document}

$\;$ \vspace{-2mm}

\noindent Read Chapter 6 of \emph{Visual Group Theory}, or Chapters
7.3, 8.1 of \emph{IBL Abstract Algebra}, and then write up solutions
to the following exercises.

\begin{enumerate}

\item Draw the subgroup lattice of the alternating group
  $A_4=\<(1\,2\,3),(1\,2)(3\,4)\>$. Then carry out the following steps
  for two of its subgroups, $H=\<(1\,2\,3)\>$ and
  $K=\<(1\,2)(3\,4)\>$. When writing a coset, list all of its
  elements.
  \begin{enumerate}
    \item Write $A_4$ as a disjoint union of the subgroup's left cosets.
    \item Write $A_4$ as a disjoint union of the subgroup's right cosets.
    \item Find all conjugates of the subgroup, and determine whether it is
      normal in $A_4$.
  \end{enumerate}



\item The \emph{center} of a group $G$ is the set
  \begin{align*}
    Z(G)=\{z\in G\mid gz=zg,\;\forall g\in G\}=\{z\in G\mid gzg^{-1}=z,\;\forall g\in G\}\,.
  \end{align*}
  \begin{enumerate}
    \item Prove that $Z(G)$ is a subgroup of $G$, and that it is normal in
  $G$. 
    \item Compute the center of the following groups: $C_6$, $D_4$, $D_5$, $Q_8$, $A_4$, $S_4$, and $D_4\times Q_8$. \medskip
  \end{enumerate}



\item The subgroup lattice of $D_4$ is shown below:
  \[
  %\xymatrixrowsep{8mm}
  %\xymatrixcolsep{4mm}
  \xymatrix{ && D_4\ar@{-}[dl]\ar@{-}[d]\ar@{-}[dr] && \\ &
    \<r^2,f\>\ar@{-}[dl]\ar@{-}[d]\ar@{-}[dr] & \<r\>\ar@{-}[d] &
    \<r^2,rf\>\ar@{-}[dl]\ar@{-}[d]\ar@{-}[dr] & \\ \<f\>\ar@{-}[drr]
    & \<r^2f\>\ar@{-}[dr] & \<r^2\>\ar@{-}[d] & \<rf\>\ar@{-}[dl] &
    \<r^3f\>\ar@{-}[dll] \\ && \<e\> && }
  \]
  For each of the 10 subgroups of $D_4$, find all of its conjugates,
  and determine whether it is normal in $D_4$. Fully justify your
  answers. [\emph{Hint}: You can do this problem without actually
    computing $xHx^{-1}$ for any subgroup $H\leq D_4$.] \medskip
  
  
  
\item Consider a chain of subgroups $K\leq H\leq G$.
  \begin{enumerate}
  \item Prove or disprove: If $K\normaleq H\normaleq G$, then
    $K\normaleq G$.
  \item Prove or disprove: If $K\normaleq G$, then $K\normaleq
    H$. \medskip
  \end{enumerate}
  
  
  
\item Let $H$ be a subgroup of $G$. Given two fixed elements $a,b\in
  G$, define the sets
    \[
    aHbH=\{ah_1bh_2\mid h_1,h_2\in H\}\,\qquad\mbox{ and }\qquad
    abH=\{abh\mid h\in H\}\,.
    \]
    Prove that if $H\normaleq G$, then $aHbH=abH$. \medskip

\item Prove that $A\times\{e_B\}$ is a normal subgroup of $A\times B$,
  where $e_B$ is the identity element of $B$. That is, show first that
  it is a subgroup, and then that it is normal. 


\end{enumerate}

\end{document}


