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\pagestyle{fancy} \lhead{{\sf Homework 8 $|$ Due Monday, October 21
}} \rhead{\thepage} \cfoot{{\sf Math 4120/6120 $|$ Modern Algebra $|$
    Fall 2019 $|$ M.~Macauley}}


\begin{document}

$\;$\vspace{-2mm}

\noindent Read Chapters 8.4--5 of \emph{Visual Group Theory}, Chapter
9.2 of \emph{IBL Abstract Algebra}, or Chapters 10.3 and 12.1 of
\emph{AATA}. Then write up solutions to the following exercises.


\begin{enumerate}

\item Let $(\Q,+)$ be the group of rational numbers under addtion,
  $(\Q^*,\cdot\,)$ the group of non-zero rational numbers under
  multiplication, and $(\Q^+,\cdot\,)$ the group of positive rational numbers
  under multiplication.
  \begin{enumerate}
    \item Show that $(\Q^*,\cdot\,)\cong(\Q^+,\cdot\,)\times
      C_2$. [\emph{Hint}: Recall that $C_2=\{e^{0\pi i},e^{\pi
          i}\}=\{1,-1\}$.]
    \item Describe the quotient groups $(\Q,+)/\<-1\>$ and
      $(\Q^*,\cdot\,)/\<-1\>$. In particular, what do the elements
      (cosets) look like?
    \item Use the Fundamental Homomorphism Theorem to prove that
      $(\Q^*,\cdot\,)/\<-1\>\cong(\Q^+,\cdot\,)$.~ \bigskip
  \end{enumerate}




  % VGT Exercise 8.12
\item For Parts (a)--(d), a group $G$ is given together with a
  normal subgroup $H$. Illustrate the embedding $\phi\colon H\to G$,
  and the quotient map $q\colon G\to G/H$, chained together so that
  $\im(\phi)=\ker(q)$. An example for $G=\Z_6$ and $H=\Z_2$ is
  shown below:
  \begin{center}
    \tikzstyle{v} = [circle, draw, fill=lightgrey,inner sep=0pt,
      minimum size=5mm] 
    \tikzstyle{v-yellow} = [circle, draw, fill=yellow,inner sep=0pt,
      minimum size=5mm] 
    \tikzstyle{v-big} = [circle, draw, fill=lightgrey,inner sep=0pt, 
      minimum size=8mm]
    \tikzstyle{v-bigy} = [circle, draw, fill=yellow,inner sep=0pt, 
      minimum size=8mm]
    \tikzstyle{to} = [draw, dashed, thick, -stealth]
    \tikzstyle{r} = [draw, very thick, Red, -stealth]
    \tikzstyle{g} = [draw, very thick, darkgreen, -stealth]
    \tikzstyle{b} = [draw, very thick, Blue]
    \begin{tikzpicture}[scale=.5,auto]
      \node (0) at (90:2) [v-yellow] {{\small $0$}};
      \node (1) at (-90:2) [v-yellow] {{\small $1$}};
      \draw [b] (0) to (1);
      %%
      \begin{scope}[shift={(9,0)}]
        \node (e) at (90:3) [v-yellow] {{\small $0$}};
        \node (r) at (30:3) [v] {{\small $1$}};
        \node (rr) at (-30:3) [v] {{\small $2$}};
        \node (rrr) at (-90:3) [v-yellow] {{\small $3$}};
        \node (rrrr) at (-150:3) [v] {{\small $4$}};
        \node (rrrrr) at (150:3) [v] {{\small $5$}};
        \path[r] (e) to [bend left=15] (r);
        \path[r] (r) to [bend left=15] (rr);
        \path[r] (rr) to [bend left=15] (rrr);
        \path[r] (rrr) to [bend left=15] (rrrr);
        \path[r] (rrrr) to [bend left=15] (rrrrr);
        \path[r] (rrrrr) to [bend left=15] (e);
      \end{scope}
    \begin{scope}[shift={(20,0)}]
      \node (00) at (90:2) [v-bigy] {{\scriptsize $\<3\>$}};
      \node (11) at (-30:2) [v-big] {{\scriptsize $1\!+\!\!\<3\>$}};
      \node (22) at (210:2) [v-big] {{\scriptsize $2\!+\!\<3\>$}};
      \draw [g] (00) to [bend left] (11);
      \draw [g] (11) to [bend left] (22);
      \draw [g] (22) to [bend left] (00);
    \end{scope}
      \draw[to] (0) to (e);
      \draw[to] (1) to (rrr);
      \draw[to] (e) to (00);
      \draw[to,bend left=10] (r) to (11);
      \draw[to] (rr) to (22);
      \draw[to,bend left=30] (rrr) to (00);
      \draw[to,bend left=25] (rrrr) to (11);
      \draw[to,bend right=15] (rrrrr) to (22);
      \node at (0,-4.5) {$H=\Z_2$};
      \node at (9,-4.5) {$H\cong\im(\phi)=\<3\><G$};
      \node at (20,-4.5) {$G/\<3\>\cong\Z_3$};
      \node at (4,3) {\small $\phi$};
      \node at (15,3) {\small $q$};
    \end{tikzpicture}
  \end{center}
  \begin{enumerate}[(a)]
  \item $G=\Z_6$, $H=\Z_3$,
  \item $G=D_3$, $H=C_3$,
  \item $G=A_4$, $H=V_4$,
  \item $G=S_n$, $H=A_n$ [don't draw the actual Cayley graphs for this
    one, just the maps].
  \end{enumerate}
  Now, answer each of the following questions about each of your
  answers to Parts (a)--(d).
  \begin{enumerate}
  \item[(e)] What map $\theta$ into $H$ would satisfy the equation
    $\im\theta=\ker\phi$? Choose one with the smallest possible domain.
  \item[(f)] What map $\theta'$ from $G/H$ would satisfy the equation
    $\im(q)=\ker(\theta')$? Choose one with the smallest possible
    codomain.
  \item[(g)] Add the two maps $\theta$ and $\theta'$ to your
    illustration.
  \item[(h)] The new chain of four homomorphisms is called a
    \emph{short exact sequence}. It is one way to use homomorphisms to
    illustrate quotients, and it shows a connection betwen embeddings
    and quotient maps. Given a normal subgroup $H\normaleq G$, show how
    to create a short exact sequence involving $G$ and $H$. 
  \end{enumerate}


\newpage

$\;$\vspace{-5mm}


\item Let $A\leq B$ and $B\normaleq G$. In this problem, you will
  prove the \emph{Diamond Isomorphism Theorem}.
  \begin{enumerate}[(a)]
  \item Show that $AB:=\{ab:a\in A,\,b\in B\}$ and $BA:=\{ba:a\in A,\,b\in B\}$ are equal as sets.
 % \item Prove set equality of $AB:=\{ab:a\in A,\,b\in B\}$ and $BA:=\{ba:a\in A,\,b\in B\}$.
 % \item Show that sets $AB:=\{ab:a\in A,\,b\in B\}$ and $BA:=\{ba:a\in A,\,b\in B\}$ are equal.
    
  \item Show that $AB$ is a subgroup of $G$.
  \item Show that $B\normaleq AB$ and $A\cap B\normaleq A$. 
  \item Show that $A/(A\cap B)\cong AB/B$. [\emph{Hint:} Construct a
    homomorphism $\phi\colon A\to AB/B$ that has kernel $A\cap B$,
    then apply the FHT.]
  \item Draw a diagram, or lattice, of $G$ and its subgroups
    $AB$, $A$, $B$, and $A\cap B$. Interpret the result in Part (c)
    in terms of this diagram. \bigskip
  \end{enumerate}


\item For each part below, consider the group $G=\<A,B\>$ generated by
  the two matrices given. Assume that matrix multiplication is the
  binary operation, and $i=\sqrt{-1}$. To what common group is $G$
  isomorphic? Write down an explicit isomorphism (you only need to
  define it for the generators), and a group presentation for $G$.
  \begin{enumerate}
    \begin{multicols}{2}
  \item $A=\begin{bmatrix}0&-1\\-1&0\end{bmatrix}$,
    $B=\begin{bmatrix}0&1\\1&0\end{bmatrix}$. 
  \item $A=\begin{bmatrix}0&-1\\-1&0\end{bmatrix}$,
    $B=\begin{bmatrix}0&i\\i&0\end{bmatrix}.$ 
  \item $A=\begin{bmatrix}0&1&0\\1&0&0\\0&0&1\end{bmatrix}$,
    $B=\begin{bmatrix}1&0&0\\0&0&1\\0&1&0\end{bmatrix}$. \bigskip
    \end{multicols}
  \end{enumerate}

    \medskip

\item In this exercise, you will prove 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 $g=a_2b_1$.]
  \item Find $\ker(\phi)$ [you need to prove your answer is correct]
    and then apply the Fundamental Homomorphism Theorem. \bigskip
  \end{enumerate}



\item For the numbers below, list all abelian groups of that order by
  writing each one as a product of cyclic groups of prime power
  order. Then, determine which group it is isomorphic to of the form
  $\Z_{n_1}\times\cdots\times\Z_{n_k}$, where $n_{i+1}\mid n_i$.

%  each one as being isomorphic to a product of
%  cyclc groups organized by ``elementary divisors,'' i.e.,

  \begin{enumerate}
    \begin{multicols}{2}
    \item $16$
    \item $54$
    \item $400$
    \item $p^2q$, where $p$ and $q$ are distinct primes.
    \end{multicols}
  \end{enumerate}

\end{enumerate}

\end{document}


