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\pagestyle{fancy} \lhead{{\sf Homework 5 $|$ Due September 24 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 4120/6120 $|$ Modern Algebra $|$
    Fall 2021 $|$ M.~Macauley}}

\begin{document}

%$\;$

\begin{enumerate}

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

\item All of the subgroups of $D_5$ should be visually apparent from
  thinking about symmetries of a regular pentagon, shown below at
  left. At right is a Cayley diagram.

  \vspace{-3mm}
  
  \colorlet{color1}{lyellow}
  \colorlet{color2}{lgreen}
  \colorlet{color3}{lred}
  \colorlet{color4}{lblue}
  \colorlet{color5}{lpurple}
  \[
  \begin{tikzpicture}[scale=1.6]
    \newdimen\r
    \r=1.1cm;
    \begin{scope}[shift={(0,0)},scale=.75]
      \coordinate (center) at (0,0);
      \draw[dashed] (18:1.25*\r)--(198:1.2*\r); 
      \draw[dashed,very thick,Blue] (90:1.25*\r)--(270:1.2*\r);
      \draw[dashed] (162:1.25*\r)--(342:1.2*\r);
      \draw[dashed] (234:1.25*\r)--(54:1.2*\r);
      \draw[dashed] (306:1.25*\r)--(126:1.2*\r);
      \draw[color1,fill=color1,opacity=.7] (0,0)--(18:1)--(90:1)--cycle;
      \draw[color2,fill=color2,opacity=.7] (0,0)--(90:1)--(162:1)--cycle;
      \draw[color3,fill=color3,opacity=.7] (0,0)--(162:1)--(234:1)--cycle;
      \draw[color4,fill=color4,opacity=.7] (0,0)--(234:1)--(306:1)--cycle;
      \draw[color5,fill=color5,opacity=.7] (0,0)--(306:1)--(18:1)--cycle;
      \draw (18:1) \foreach \x in {90,162,234,306} { -- (\x:1) }
      -- cycle (center);
      \draw[-stealth',Red] (1.3,-.8) to[very thick,bend right=60] (1.5,-.1);
      \node at (1.75,-.5) {\Alert{$r$}};
      \node at (-.2,1.3) {\Balert{$f$}};
      \node at (54:.5) {\large \textbf{3}};
      \node at (126:.5) {\large \textbf{4}};
      \node at (198:.5) {\large \textbf{5}};
      \node at (270:.5) {\large \textbf{1}};
      \node at (342:.5) {\large \textbf{2}};
    \end{scope}
    %%
    \begin{scope}[shift={(4,0)},scale=.35]
      \tikzstyle{every node}=[font=\footnotesize]
      \tikzstyle{r-out} = [draw, very thick, Red,-stealth,bend right=22]
      \tikzstyle{r-in} = [draw, very thick, Red,-stealth,bend left=20]
      \node (e) at (0:4) [v] {$1$};
      \node (r) at (72:4) [v] {$r$};
      \node (r2) at (144:4) [v] {$r^2$};
      \node (r3) at (216:4) [v] {$r^3$};
      \node (r4) at (288:4) [v] {$r^4$};
      \node (f) at (0:2) [v] {$f$};
      \node (rf) at (72:2) [v] {$rf$};
      \node (r2f) at (144:2) [v] {$r^2\!f$};
      \node (r3f) at (216:2) [v] {$r^3\!f$};
      \node (r4f) at (288:2) [v] {$r^4\!f$};
      \draw [r-out] (e) 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 (e);
      \draw [r-in] (f) to (r4f);
      \draw [r-in] (r4f) to (r3f);
      \draw [r-in] (r3f) to (r2f);
      \draw [r-in] (r2f) to (rf);
      \draw [r-in] (rf) to (f);
      \draw [bb] (e) to (f);
      \draw [bb] (r) to (rf);
      \draw [bb] (r2) to (r2f);
      \draw [bb] (r3) to (r3f);
      \draw [bb] (r4) to (r4f);
    \end{scope}
  \end{tikzpicture}
  \]

  \begin{enumerate}
  \item Construct a subgroup lattice for $D_5$. Label each edge
    from $H$ to $K$ with $[H:K]$.
  \item Find the left and right cosets of the subgroups $\<f\>$ and
    $\<r\>$, and find their normalizers.
  \item Two subgroups $H,K\leq G$ are \emph{conjugate} if
    $K=gHg^{-1}:=\{ghg^{-1}\mid h\in H\}$ for some $g\in G$. This
    defines an equivalence relation on the set of subgroups called
    \emph{conjugacy classes}. Partition the subgroups of $D_5$ into
    conjugacy classes.
  %\item Repeat the previous part for $D_4$. The answer will be
  %  different because $4$ is even.
  \end{enumerate}
  
  %%-------------------------------------------------------------------

  %%-------------------------------------------------------------------
  
\item The Cayley diagram of the smallest non-abelian group, $G=C_7\rtimes C_3$, is shown below. 

  \tikzstyle{R1} = [draw, very thick, Red,-stealth,bend right=15]
\tikzstyle{R2} = [draw, very thick, Red,-stealth,bend right=20]
\tikzstyle{v-tiny} = [circle, draw, fill=lightgrey,inner sep=0pt,
  minimum size=6mm]
\tikzstyle{v-small} = [circle, draw, fill=lightgrey,inner sep=0pt,
  minimum size=6mm]
\tikzstyle{v-big} = [circle, draw, fill=lightgrey,inner sep=0pt,
  minimum size=6mm]
\[
  \begin{tikzpicture}[scale=1.25]
   \tikzstyle{every node}=[font=\footnotesize]
    \begin{scope}[shift={(0,0)},scale=1]
      \node (s2) at (0:1) [v-small] {$s^2$};
      \node (rs2) at (51.45:1) [v-small] {$rs^2$};
      \node (r2s2) at (102.9:1) [v-small] {$r^2\!s^2$};
      \node (r3s2) at (154.3:1) [v-small] {$r^3\!s^2$};
      \node (r4s2) at (205.7:1) [v-small] {$r^4\!s^2$};
      \node (r5s2) at (257.1:1) [v-small] {$r^5\!s^2$};
      \node (r6s2) at (308.6:1) [v-small] {$r^6\!s^2$};
      %%
      \node (s) at (0:2) [v-small] {$s$};
      \node (rs) at (51.45:2) [v-small] {$rs$};
      \node (r2s) at (102.9:2) [v-small] {$r^2\!s$};
      \node (r3s) at (154.3:2) [v-small] {$r^3\!s$};
      \node (r4s) at (205.7:2) [v-small] {$r^4\!s$};
      \node (r5s) at (257.1:2) [v-small] {$r^5\!s$};
      \node (r6s) at (308.6:2) [v-small] {$r^6\!s$};
      %%
      \node (1) at (0:3) [v-small] {$1$};
      \node (r) at (51.45:3) [v-small] {$r$};
      \node (r2) at (102.9:3) [v-small] {$r^2$};
      \node (r3) at (154.3:3) [v-small] {$r^3$};
      \node (r4) at (205.7:3) [v-small] {$r^4$};
      \node (r5) at (257.1:3) [v-small] {$r^5$};
      \node (r6) at (308.6:3) [v-small] {$r^6$};
      %%
      \draw [R1] (1) to (r);
      \draw [R1] (r) to (r2);
      \draw [R1] (r2) to (r3);
      \draw [R1] (r3) to (r4);
      \draw [R1] (r4) to (r5);
      \draw [R1] (r5) to (r6);
      \draw [R1] (r6) to (1);
      %%
      \draw [R2] (s) to (r2s);
      \draw [R2] (r2s) to (r4s);
      \draw [R2] (r4s) to (r6s);
      \draw [R2] (r6s) to (rs);
      \draw [R2] (rs) to (r3s);
      \draw [R2] (r3s) to (r5s);
      \draw [R2] (r5s) to (s);
      %%
      \draw [r] (s2) to (r4s2);
      \draw [r] (r4s2) to (rs2);
      \draw [r] (rs2) to (r5s2);
      \draw [r] (r5s2) to (r2s2);
      \draw [r] (r2s2) to (r6s2);
      \draw [r] (r6s2) to (r3s2);
      \draw [r] (r3s2) to (s2);
      %%
      \draw [b] (1) to (s); \draw [b] (s) to (s2);
      \draw [b,bend left] (s2) to (1);
      \draw [b] (r) to (rs); \draw [b] (rs) to (rs2);
      \draw [b,bend left] (rs2) to (r);
      \draw [b] (r2) to (r2s); \draw [b] (r2s) to (r2s2);
      \draw [b,bend left] (r2s2) to (r2);
      \draw [b] (r3) to (r3s); \draw [b] (r3s) to (r3s2);
      \draw [b,bend left] (r3s2) to (r3);
      \draw [b] (r4) to (r4s); \draw [b] (r4s) to (r4s2);
      \draw [b,bend left] (r4s2) to (r4);
      \draw [b] (r5) to (r5s); \draw [b] (r5s) to (r5s2);
      \draw [b,bend left] (r5s2) to (r5);
      \draw [b] (r6) to (r6s); \draw [b] (r6s) to (r6s2);
      \draw [b,bend left] (r6s2) to (r6);
    \end{scope}
    \end{tikzpicture}
%    \caption{The Cayley diagram of the semidirect product $C_7\rtimes_\theta C_3$, the smallest nonabelian group of odd order.}\label{fig2:C7xC3-semi}
  \]
  \begin{enumerate}
  \item On a blank Cayley diagram, label nodes with the order of
    the corresponding elements.
  \item Construct a subgroup lattice and label each edge with the
    corresponding index.
  \item Find the left and right cosets of the subgroups $\<s\>$ and
    $\<r\>$, and their normalizers.
  \item Partition the subgroups into conjugacy classes, and denote
    this on your lattice.
  \item Repeat the previous parts for the group $G=C_9\rtimes C_3$,
    whose Cayley diagram you constructed on the previous
    assignment. It is helpful to know that it has four subgroups of
    order $9$ and four subgroups of order $3$.
  \end{enumerate}

  %%-------------------------------------------------------------------
  
\item Consider two semidirect products of $C_5$ with $C_4$, whose
  Cayley diagrams are shown below.

  \vspace{-3mm}

  \tikzstyle{v-small} = [circle, draw, fill=lightgrey,inner sep=0pt, 
    minimum size=6mm]
  \tikzstyle{B2} = [draw, Blue, -stealth']       % Blue --->
  \[
  \hspace*{-2em}
  \begin{tikzpicture}[scale=1]
    %%%%
    %%%%%%%%%%%%
    %%%%%%%%%%%%
    %%%%%%%%%%%%
    \begin{scope}[shift={(0,0)}]
      \tikzstyle{every node}=[font=\scriptsize]
      \node at (2.25,2.5) {\normalsize $\Alert{C_5}\!\rtimes_{\theta_1}\!\!\Balert{C_4}$};
      \begin{scope}[shift={(0,4.5)}]
        \node (1) at (0:1) [v-small] {$1$};
        \node (r) at (72:1) [v-small] {$a$};
        \node (r2) at (144:1) [v-small] {$a^2$};
        \node (r3) at (216:1) [v-small] {$a^3$};
        \node (r4) at (288:1) [v-small] {$a^4$};
      \end{scope}
      \begin{scope}[shift={(0,0)}]
        \node (s) at (0:1) [v-small] {$b$};
        \node (rs) at (72:1) [v-small] {$ab$};
        \node (r2s) at (144:1) [v-small] {$a^2\!b$};
        \node (r3s) at (216:1) [v-small] {$a^3\!b$};
        \node (r4s) at (288:1) [v-small] {$a^4\!b$};
      \end{scope}
      \begin{scope}[shift={(5,0)}]
        \node (s2) at (0:1) [v-small] {$b^2$};
        \node (rs2) at (72:1) [v-small] {$ab^2$};
        \node (r2s2) at (144:1) [v-small] {$a^2\!b^{2}$};
        \node (r3s2) at (216:1) [v-small] {$a^3\!b^{2}$};
        \node (r4s2) at (288:1) [v-small] {$a^4\!b^{2}$};
      \end{scope}
      \begin{scope}[shift={(5,4.5)}]
        \node (s3) at (0:1) [v-small] {$b^3$};
        \node (rs3) at (72:1) [v-small] {$ab^{\!3}$};
        \node (r2s3) at (144:1) [v-small] {$a^2\!b^3$};
        \node (r3s3) at (216:1) [v-small] {$a^3\!b^{3}$};
        \node (r4s3) at (288:1) [v-small] {$a^4\!b^{3}$};
      \end{scope}
      \draw [B2] (1) to (s);
      \draw [B2] (r) to[bend left=18] (rs);
      \draw [B2] (r2) to[bend right=21] (r2s); 
      \draw [B2] (r3) to[bend left=22] (r3s);
      \draw [B2] (r4) to[bend right=18] (r4s);
      %%
      \draw [B2] (s) to (s2);
      \draw [B2] (rs) to (rs2);
      \draw [B2] (r2s) to (r2s2);
      \draw [B2] (r3s) to (r3s2);
      \draw [B2] (r4s) to (r4s2);
      %%
      \draw [B2] (s2) to (s3);
      \draw [B2] (rs2) to[bend right=18] (rs3);
      \draw [B2] (r2s2) to[bend left=21] (r2s3);
      \draw [B2] (r3s2) to[bend right=22] (r3s3);
      \draw [B2] (r4s2) to[bend left=18] (r4s3);
      %%
      \draw [B2] (s3) to (1); 
      \draw [B2] (rs3) to (r);
      \draw [B2] (r2s3) to (r2);
      \draw [B2] (r3s3) to (r3);
      \draw [B2] (r4s3) to (r4);
    %%
      \draw [r] (1) to (r); \draw [r] (r) to (r2); \draw [r] (r2) to (r3);
      \draw [r] (r3) to (r4); \draw [r] (r4) to (1);
      %%
      \draw [r] (s) to (r2s); \draw [r] (r2s) to (r4s); \draw [r] (r4s) to (rs);
      \draw [r] (rs) to (r3s); \draw [r] (r3s) to (s);
      %%
      \draw [r] (s2) to (r4s2); \draw [r] (r4s2) to (r3s2);
      \draw [r] (r3s2) to (r2s2); \draw [r] (r2s2) to (rs2);
      \draw [r] (rs2) to (s2);
      %%
      \draw [r] (s3) to (r3s3); \draw [r] (r3s3) to (rs3);
      \draw [r] (rs3) to (r4s3); \draw [r] (r4s3) to (r2s3);
      \draw [r] (r2s3) to (s3);
    \end{scope}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
        \begin{scope}[shift={(9,0)}]
      \tikzstyle{every node}=[font=\scriptsize]
      \node at (2.35,2.5) {\normalsize $\Alert{C_5}\!\rtimes_{\theta_2}\!\!\Balert{C_4}$};
      \begin{scope}[shift={(0,4.5)}]
        \node (1) at (0:1) [v-small] {$1$};
        \node (r) at (72:1) [v-small] {$a$};
        \node (r2) at (144:1) [v-small] {$a^2$};
        \node (r3) at (216:1) [v-small] {$a^3$};
        \node (r4) at (288:1) [v-small] {$a^4$};
      \end{scope}
      \begin{scope}[shift={(0,0)}]
        \node (s) at (0:1) [v-small] {$b$};
        \node (rs) at (72:1) [v-small] {$ab$};
        \node (r2s) at (144:1) [v-small] {$a^2\!b$};
        \node (r3s) at (216:1) [v-small] {$a^3\!b$};
        \node (r4s) at (288:1) [v-small] {$a^4\!b$};
      \end{scope}
      \begin{scope}[shift={(5,0)}]
        \node (s2) at (0:1) [v-small] {$b^2$};
        \node (rs2) at (72:1) [v-small] {$ab^2$};
        \node (r2s2) at (144:1) [v-small] {$a^2\!b^{2}$};
        \node (r3s2) at (216:1) [v-small] {$a^3\!b^{2}$};
        \node (r4s2) at (288:1) [v-small] {$a^4\!b^{2}$};
      \end{scope}
      \begin{scope}[shift={(5,4.5)}]
        \node (s3) at (0:1) [v-small] {$b^3$};
        \node (rs3) at (72:1) [v-small] {$ab^{\!3}$};
        \node (r2s3) at (144:1) [v-small] {$a^2\!b^{3}$};
        \node (r3s3) at (216:1) [v-small] {$a^3\!b^{3}$};
        \node (r4s3) at (288:1) [v-small] {$a^4\!b^{3}$};
      \end{scope}
      \draw [B2] (1) to (s);
      \draw [B2] (r) to[bend left=18] (rs);
      \draw [B2] (r2) to[bend right=21] (r2s); 
      \draw [B2] (r3) to[bend left=22] (r3s);
      \draw [B2] (r4) to[bend right=18] (r4s);
      %%
      \draw [B2] (s) to (s2);
      \draw [B2] (rs) to (rs2);
      \draw [B2] (r2s) to (r2s2);
      \draw [B2] (r3s) to (r3s2);
      \draw [B2] (r4s) to (r4s2);
      %%
      \draw [B2] (s2) to (s3);
      \draw [B2] (rs2) to[bend right=18] (rs3);
      \draw [B2] (r2s2) to[bend left=21] (r2s3);
      \draw [B2] (r3s2) to[bend right=22] (r3s3);
      \draw [B2] (r4s2) to[bend left=18] (r4s3);
      %%
      \draw [B2] (s3) to (1); 
      \draw [B2] (rs3) to (r);
      \draw [B2] (r2s3) to (r2);
      \draw [B2] (r3s3) to (r3);
      \draw [B2] (r4s3) to (r4);
      %%
      \draw [r] (1) to (r); \draw [r] (r) to (r2); \draw [r] (r2) to (r3);
      \draw [r] (r3) to (r4); \draw [r] (r4) to (1);
      %%
      \draw [r] (s) to (r4s); \draw [r] (r4s) to (r3s);
      \draw [r] (r3s) to (r2s); \draw [r] (r2s) to (rs);
      \draw [r] (rs) to (s);
      %%
      \draw [r] (s2) to (rs2); \draw [r] (rs2) to (r2s2); \draw [r] (r2s2) to (r3s2);
      \draw [r] (r3s2) to (r4s2); \draw [r] (r4s2) to (s2);
      %%
      \draw [r] (s3) to (r4s3); \draw [r] (r4s3) to (r3s3);
      \draw [r] (r3s3) to (r2s3); \draw [r] (r2s3) to (rs3);
      \draw [r] (rs3) to (s3); 
   \end{scope}
    \end{tikzpicture}
\]

\begin{enumerate}[(a)]
\item On blank Cayley diagrams, label the order of each element. 
\item The subgroup lattices of these two groups are shown below, not
  necessarily in the right order. Determine which lattice corresponds
  to which Cayley diagram (with justification), and then re-draw them
  with the subgroups written by generators.
  \[
  \hspace*{-3em}
  \begin{tikzpicture}[shorten >= -2pt, shorten <= -2pt,scale=1.4]
    \tikzstyle{every node}=[font=\small]
    \begin{scope}[shift={(0,-1.3)},scale=1.3]
      \node (G) at (0,4) {$G$};
      \node(s2-t) at (-.15,2.9) {$D_5$};
      \node (t) at (-1.4,2.2) {$C_5$};
      \node (s) at (-.7,2) {$C_4$};
      \node (st) at (.25,2) {$C_4$};
      \node (s3t) at (1.15,2) {$C_4$};
      \node (ts) at (1.8,2) {$C_4$};
      \node (ts3) at (2.5,2) {$C_4$};
      \node(s2) at (-.45,1.1) {$C_2$}; 
      \node(ts2) at (.25,1.1) {$C_2$}; 
      \node(sts) at (1.15,1.1) {$C_2$}; 
      \node(s2t2) at (1.8,1.1) {$C_2$};
      \node(s2t) at (2.5,1.1) {$C_2$};
      \node (1) at (0,0) {$\<1\>$};
      \draw[bend right=5] (G) to (s); 
      \draw (G) to (st); \draw (G) to (s3t);
      \draw (G) to (ts); \draw (G) to (ts3); \draw (G) to (s2-t);
      %%
      \draw (s2-t) to (t); \draw (t) to (1);
      \draw (s2-t) to (s2); \draw (s2-t) to (ts2);
      \draw (s2-t) to (sts);
      \draw (s2-t) to (s2t2); 
      \draw[bend left=10] (s2-t) to (s2t);
      %%
      \draw (s) to (s2); \draw (st) to (ts2); \draw (s3t) to (sts);
      \draw (ts) to (s2t2); \draw (ts3) to (s2t); 
      %%
      \draw (1) to (s2); \draw (1) to (ts2); \draw (1) to (sts);
      \draw (1) to (s2t2); \draw (1) to (s2t);
    \end{scope}
    %%
    \begin{scope}[shift={(6.75,0)},scale=1.3]
      \node(G) at (0,3) {$G$};
      \node(r) at (-1.5,2) {$C_{10}$};
      \node(s) at (-.5,1.1) {$C_4$}; 
      \node(sr) at (.15,1.1) {$C_4$}; 
      \node(sr2) at (.8,1.1) {$C_4$}; 
      \node(r2s) at (1.45,1.1) {$C_4$};
      \node(rs) at (2.1,1.1) {$C_4$};
      \node (r5) at (0,0) {$C_2$};
      \node (r2) at (-1.75,1.3) {$C_5$};
      \node (1) at (0,-1) {$\<1\>$};
      \draw (G) to (r); \draw (r) to (r5);
      \draw (G) to (s); \draw (G) to (sr); 
      \draw (G) to (sr2); \draw (G) to (r2s); \draw (G) to (rs);
      \draw (r5) to (s); \draw (r5) to (sr); \draw (r5) to (sr2);
      \draw (r5) to (r2s); \draw (r5) to (rs);
      \draw (1) to (r5); \draw (1) to (r2); \draw (r) to (r2); 
    \end{scope}
\end{tikzpicture}
  \]
  
\item Determine which group each of these is isomorphic to, and which elements $a$ and $b$ correspond to. Recall that there are only three nonabelian groups of order $20$:
  \[
  D_{10}=\big\<r,f\mid r^{10}=f^2=1,\,rfr=f\big\>,\qquad \Dic_{10}=\big\<r,s\mid r^{10}=s^4=1,\,r^5=s^2\big\>,
  \]
  \[
  \AGL_1(\Z_5)=\left\<\begin{bmatrix}1&1\\0&1\end{bmatrix},\,
 \begin{bmatrix}2&1\\0&1\end{bmatrix}\right\>\leq\GL_2(\Z_5).
   \]
   Write a presentation for both groups in this problem, in terms of
   $a$ and $b$.
  
\end{enumerate}

%\newpage  

  %%-------------------------------------------------------------------
  
\item Prove the following, algebraically (that is, do not refer to
  Cayley diagrams):
  \begin{enumerate}[(a)]
  \item If $\mathcal{H}$ is a collection of subgroups of $G$, then
    $\displaystyle\bigcap_{H_\alpha\in\mathcal{H}}H_\alpha$ is a subgroup of $G$.
  \item For any (possibly infinite) subset $S\subseteq G$, the
    subgroup generated by $S$ is defined as
    \[
    \<S\>:=\big\{s_1^{e_1}s_2^{e_2}\cdots s_k^{e_k}\mid s_i\in
    S,\;e_i\in\{-1,1\}\big\}.
    \]
    That is, $\<S\>$ consists of all finite ``words'' that can be
    written using the elements in $S$ and their inverses. Note that
    the $s_i$'s need not be distinct. Prove that
    \[
    \<S\>=\bigcap_{S\subseteq H_\alpha\leq G} H_\alpha,
    \]
    where the intersection is taken over all subgroups of $G$ that
    contain $S$. [\emph{Hint}: To prove that $A=B$, you need to show that
      that $A\subseteq B$ and $B\subseteq A$.]
  \end{enumerate}
  
  %%-------------------------------------------------------------------

 \end{enumerate}

\end{document}








