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\pagestyle{fancy} \lhead{{\sf Homework 8 $|$ Due March 7 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 4120/6120 $|$ Visual Algebra $|$
    Spring 2025 $|$ M.~Macauley}}

\begin{document}

$\;$

\begin{enumerate}

  %%-------------------------------------------------------------------

  \item Let $G$ be the \emph{semiabelian group} of order 16, defined by
  the presentation
  \[
  \SA_8=\big\<r,s\mid r^8=s^2=1, srs=r^5\big\>,
  \]
  A Cayley graph and subgroup lattice are shown below.

  \vspace{-7mm}
  
  \[
  \hspace*{-4mm}
  \begin{tikzpicture}[scale=1.55]
    \tikzstyle{R-in} = [draw, very thick, eRed,-stealth,bend right=12]
    \tikzstyle{R-out} = [draw, very thick, eRed,-stealth,bend right=15]
    %%
    \begin{scope}[shift={(0,0)},scale=1]
      \tikzstyle{every node}=[font=\footnotesize]
      \node (s) at (0:1.1) [v] {$s$};
      \node (rs) at (45:1.15) [v] {$rs$};
      \node (r2s) at (90:1.15) [v] {$r^2\!s$};
      \node (r3s) at (135:1.15) [v] {$r^3\!s$};
      \node (r4s) at (180:1.15) [v] {$r^4\!s$};
      \node (r5s) at (225:1.15) [v] {$r^5\!s$};
      \node (r6s) at (270:1.15) [v] {$r^6\!s$};
      \node (r7s) at (315:1.15) [v] {$r^7\!s$};
      %%
      \node (1) at (0:2) [v] {$1$};
      \node (r) at (45:2) [v] {$r$};
      \node (r2) at (90:2) [v] {$r^2$};
      \node (r3) at (135:2) [v] {$r^3$};
      \node (r4) at (180:2) [v] {$r^4$};
      \node (r5) at (225:2) [v] {$r^5$};
      \node (r6) at (270:2) [v] {$r^6$};
      \node (r7) at (315:2) [v] {$r^7$};
      %%
      \draw [R-out] (1) to (r);
      \draw [R-out] (r) to (r2);
      \draw [R-out] (r2) to (r3);
      \draw [R-out] (r3) to (r4);
      \draw [R-out] (r4) to (r5);
      \draw [R-out] (r5) to (r6);
      \draw [R-out] (r6) to (r7);
      \draw [R-out] (r7) to (1);
      %%
      \draw [r] (s) to (r5s);
      \draw [r] (r5s) to (r2s);
      \draw [r] (r2s) to (r7s);
      \draw [r] (r7s) to (r4s);
      \draw [r] (r4s) to (rs);
      \draw [r] (rs) to (r6s);
      \draw [r] (r6s) to (r3s);
      \draw [r] (r3s) to (s);
      %%
      \draw [bb] (1) to (s); \draw [bb] (r) to (rs);
      \draw [bb] (r2) to (r2s); \draw [bb] (r3) to (r3s);
      \draw [bb] (r4) to (r4s); \draw [bb] (r5) to (r5s);
      \draw [bb] (r6) to (r6s); \draw [bb] (r7) to (r7s);
      %%
      \node at (0,0) {\normalsize $\SA_8$}; 
    \end{scope}
    %%
    \begin{scope}[shift={(4.25,-2.15)},shorten >= -2pt, shorten <= -2pt,
        scale=.7]
      \tikzstyle{every node}=[font=\small]
      %%
      \node(G) at (3.5,6) {$\<r,s\>$};
      \node(rs) at (3.5,4.5) {$\<rs\>$};
      \node(r2-s) at (1.75,4.5) {$\<r^2,s\>$};
      \node(r) at (5.25,4.5) {$\<r\>$};
      \node(r4-s) at (0,3) {$\<r^4,s\>$};
      \node(r2s) at (1.75,3) {$\<r^2s\>$};
      \node(r2) at (3.5,3) {$\<r^2\>$};
      \node(s) at (-1.75,1.5) {$\<s\>$};
      \node(r4s) at (0,1.5) {$\<r^4s\>$};
      \node(r4) at (1.75,1.5) {$\<r^4\>$};
      \node(1) at (0,0) {$\<1\>$};
      \draw[f] (1) to (s);
      \draw[f] (1) to (r4s);
      \draw[f] (1) to (r4);
      \draw[f] (s) to (r4-s);
      \draw[f] (r4s) to (r4-s);
      \draw[f] (r4) to (r4-s);
      \draw[f] (r4) to (r2s);
      \draw[f] (r4) to (r2);
      \draw[f] (r4-s) to (r2-s);
      \draw[f] (r2s) to (r2-s);
      \draw[f] (r2) to (r2-s);
      \draw[f] (r2) to (r);
      \draw[f] (r2) to (rs);
      \draw[f] (r2-s) to (G);
      \draw[f] (r) to (G);
      \draw[f] (rs) to (G);
    \end{scope}
  \end{tikzpicture}
  \]
  \begin{enumerate}
  \item The subgroups $V=\<r^4,s\>$, $H=\<r^2s\>$, $K=\<r^2\>$, and
    $N=\<r^4\>$ are all normal. Highlight their cosets on a fresh Cayley
    diagram by colors.
  \item Construct a Cayley table for the quotient of $G$ by each of
    these subgroups. Then draw a Cayley graph for each, labeling the
    nodes with elements (i.e., cosets).
  \item Let $N=\<r^4\>$. The shaded region below shows an order-$4$
    cyclic subgroup of $G/N$, generated by the element
    $rN$, and how the union of these four cosets is the
    order-$8$ subgroup $\<r\>$ of $G$. Construct analogous tables for
    the other five non-trivial proper subgroups of $G/N$, and
    then draw the subgroup lattice of $G/N$.
    
    \vspace{-2mm}
    
    \[
    \hspace*{-5mm}
    \begin{tikzpicture}[xscale=.9]
      \tikzstyle{every node}=[font=\small,anchor=south]
      %%
      \begin{scope}[shift={(0,0)}]
        \draw[thin,fill=faded] (0,0) rectangle (2,3.2);
        \draw[thin] (0,2.4) to (4,2.4); 
        \draw[thin] (0,1.6) to (4,1.6);
        \draw[thin] (0,.8) to (4,.8);
        \draw[thin] (2,0) to (4,0);
        \draw[thin] (2,3.2) to (4,3.2);
        \draw[thin] (4,0) to (4,3.2);
        \node at (1,.1) {$N$};
        \node at (3,.1) {$sN$};
        \node at (1,.9) {$rN$};
        \node at (3,.9) {$rsN$};
        \node at (1,1.7) {$r^2N$};
        \node at (3,1.7) {$r^2sN$};
        \node at (1,2.5) {$r^3N$};
        \node at (3,2.5) {$r^3sN$};
        \node at (2,-.8) {$\<rN\>\leq G/N$};
      \end{scope}
      %%
      \begin{scope}[shift={(6,0)}]
        \draw[thin,fill=faded] (0,0) rectangle (2,3.2);
        \draw[thin] (0,2.4) to (4,2.4); 
        \draw[thin] (0,1.6) to (4,1.6);
        \draw[thin] (0,.8) to (4,.8);
        \draw[thin] (2,0) to (4,0);
        \draw[thin] (2,3.2) to (4,3.2);
        \draw[thin] (4,0) to (4,3.2);
        \node at (.5,.15) {$1$};
        \node at (1.5,.15) {$r^4$};
        \node at (2.5,.15) {$s$};
        \node at (3.5,.15) {$r^4s$};
        \node at (.5,.95) {$r$};
        \node at (1.5,.95) {$r^5$};
        \node at (2.5,.95) {$rs$};
        \node at (3.5,.95) {$r^5s$};
        \node at (.5,1.75) {$r^2$};
        \node at (1.5,1.75) {$r^6$};
        \node at (2.5,1.75) {$r^2s$};
        \node at (3.5,1.75) {$r^6s$};
        \node at (.5,2.55) {$r^3$};
        \node at (1.5,2.55) {$r^7$};
        \node at (2.5,2.55) {$r^3s$};
        \node at (3.5,2.55) {$r^7s$};
        \node at (2,-.8) {$\<r\>/N\leq G/N$};
      \end{scope}
      %%
      \begin{scope}[shift={(12,0)}]
        \draw[thin,fill=faded] (0,0) rectangle (2,3.2);
        \draw[thin] (2,0) to (4,0);
        \draw[thin] (2,3.2) to (4,3.2);
        \draw[thin] (4,0) to (4,3.2);
        \node at (.5,.15) {$1$};
        \node at (1.5,.15) {$r^4$};
        \node at (2.5,.15) {$s$};
        \node at (3.5,.15) {$r^4s$};
        \node at (.5,.95) {$r$};
        \node at (1.5,.95) {$r^5$};
        \node at (2.5,.95) {$rs$};
        \node at (3.5,.95) {$r^5s$};
        \node at (.5,1.75) {$r^2$};
        \node at (1.5,1.75) {$r^6$};
        \node at (2.5,1.75) {$r^2s$};
        \node at (3.5,1.75) {$r^6s$};
        \node at (.5,2.55) {$r^3$};
        \node at (1.5,2.55) {$r^7$};
        \node at (2.5,2.55) {$r^3s$};
        \node at (3.5,2.55) {$r^7s$};
        \node at (2,-.8) {$\<r\>\leq G$};
      \end{scope}
    \end{tikzpicture}
    \] \vspace{-9mm}
    
  \item For each subgroup $M/N$ from Part~(c), determine what the
    quotient of $G/N$ (order $8$) by $M/N$ (order $4$ or $2$) is
    isomorphic to. Justify your answer.
    
  \item One step of Part~(c) consisted of starting with $G$, taking
    the quotient by $N$, and then taking the subgroup generated by
    $r^2N$ and $sN$.  Compare and contrast this to doing these steps
    in the reverse order. That is: \emph{start with $G$, first take
      the subgroup $\<r^2,s\>$, and then take the quotient by $N$}.  
    
  \item Repeat Part~(c) for subgroups $V=\<r^4,s\>$, $H=\<r^2s\>$, and
    $K=\<r^2\>$ of $G$. This time, include the trivial and proper
    subgroups for each.
    
  \end{enumerate}
  
  \newpage
  
  %%====================================================================
  
\item Show that there is no embedding $\phi\colon\Z_n\into\Z$. \bigskip
  
  %%====================================================================
  
\item All of the following can be done by defining an explicit map,
  showing that it is a homomorphism, and a bijection.
  \begin{enumerate}
  \item Show that $A\times B\cong B\times A$.
  \item Show that $xHx^{-1}\cong H$, for any  $H\leq G$.
    Conclude that $|xy|=|yx|$ for any $x,y\in G$.
  \item Show that $(\Q^*,\cdot\,)\cong(\Q^+,\cdot\,)\times
    C_2$. Recall that $\Q^*$ and $\Q^+$ are the nonzero and positive
    rational numbers, respectively, and $C_2=\{1,-1\}$. \bigskip
  \end{enumerate}
  
  %%====================================================================
  
\item Let $\phi\colon G\to H$ be a homomorphism, and $N\normaleq H$. 
  \begin{enumerate}
  \item Show that the set $\phi^{-1}(N):=\big\{g\in G\mid\phi(g)\in N\big\}$
    is a subgroup of $G$. 
  \item Show that $\phi^{-1}(N)$ is a \emph{normal} subgroup of $G$.
  \item Show by example that if $M\normaleq G$, then $\phi(M)$ need not
    be a normal subgroup of $H$. \bigskip
  \end{enumerate}
  
  %%====================================================================
  
\item In this exercise, you will show that if $A$ and $B$ are normal
  subgroups and $AB=G$, then
  \[
  G/(A\cap B)\cong(G/A)\times(G/B).
  \]
  \begin{enumerate}
  \item Consider the following map:
    \[
    \phi\colon AB\longto(G/A)\times(G/B)\,,\qquad \phi(g)=(gA,gB).
    \]
    Show that $\phi$ is a homomorphism.
  \item Show that $\phi$ is surjective. That is, given any
    $(g_1A,g_2B)$, show that there is some $g=ab\in AB$ such that
    $\phi(g)=(g_1A,g_2B)$. [\emph{Hint}: Try $g=a_2b_1$; show this works.]
  \item Find $\Ker(\phi)$ [you need to verify your answer is correct]
    and then apply the fundamental homomorphism theorem. \bigskip
  \end{enumerate}

  %%---------------------------------------------------------------
  
\end{enumerate}

\end{document}    
    
  
 
