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\pagestyle{fancy} \lhead{{\sf Homework 2 $|$ Due September 8 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 8510 $|$ Abstract Algebra $|$
    Fall 2023 $|$ M.~Macauley}}

\begin{document}

$\;$

\begin{enumerate}
  
  %%-------------------------------------------------------------------

\item Cayley graphs of two groups are shown below. Carry out the
  following steps for each group.
  \begin{enumerate}
  \item Find this group on the LMFDB
    (\url{https://beta.lmfdb.org/Groups/}). What is its label, and
    what other group(s) is it isomorphic to?
  \item Construct a cycle graph, and label the nodes with group elements. 
  \item Write a presentation using the generators in the Cayley graph. 
  \item Find the left and right cosets of the following subgroups
    $H=\<r\>$, $K=\<s\>$, $L=\<r^3\>$.
  \item Find the normalizers of $H$, $K$, and $L$. Write them first as a
    union of cosets, and then as a subgroup by generator(s). Determine
    the isomorphism type of each.
  \item Find all conjugate subgroups of $H$, $K$, and $L$. Write each
    group by generator.
  \item Construct the subgroup lattice, and write each subgroup with a
    minimal generating set. Denote the conjugacy classes of subgroups by
    dashed circles.
  \item What is the center, $Z(G)$?
  \end{enumerate}

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

  \tikzstyle{R9} = [draw, very thick, Red,-stealth,bend right=10]
  \tikzstyle{R3} = [draw, very thick, Red,-stealth,bend left=8]
  \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=5.5mm]
  %%
   \[
  \hspace*{-12mm}
  \begin{tikzpicture}[scale=1.25]
    \tikzstyle{every node}=[font=\scriptsize]
   \begin{scope}[scale=1,shift={(0,0)}]
      \node (e0) at (3,5) [v] {$1$};
      \node (f1) at (3,4) [v] {$r$};
      \node (e2) at (3,3) [v] {$r^2$};
      \node (f0) at (3,2) [v] {$r^3$};
      \node (e1) at (3,1) [v] {$r^4$};
      \node (f2) at (3,0) [v] {$r^5$};
      %%
      \node (r0) at (1.5,5) [v] {$s$};
      \node (rf1) at (1.5,4) [v] {$sr$};
      \node (r2) at (1.5,3) [v] {$sr^2$};
      \node (rf0) at (1.5,2) [v] {$sr^3$};
      \node (r1) at (1.5,1) [v] {$sr^4$};
      \node (rf2) at (1.5,0) [v] {$sr^5$};
      %%
      \node (rr0) at (0,5) [v] {$s^2$}; 
      \node (rrf1) at (0,4) [v] {$s^2r$};
      \node (rr2) at (0,3) [v] {$s^2\!r^2$};
      \node (rrf0) at (0,2) [v] {$s^2\!r^3$};
      \node (rr1) at (0,1) [v] {$s^2\!r^4$};
      \node (rrf2) at (0,0) [v] {$s^2\!r^5$};
      %%
      \draw [r] (e0) to (f1);
      \draw [r] (f1) to (e2);
      \draw [r] (e2) to (f0);
      \draw [r] (f0) to (e1);
      \draw [r] (e1) to (f2);      
      \draw [r] (f2) to [bend right=22] (e0);
      %%
      \draw [r] (r0) to (rf1);
      \draw [r] (rf1) to (r2);
      \draw [r] (r2) to (rf0);
      \draw [r] (rf0) to (r1);
      \draw [r] (r1) to (rf2);      
      \draw [r] (rf2) to [bend right=22] (r0);
      %%
      \draw [r] (rr0) to (rrf1);
      \draw [r] (rrf1) to (rr2);
      \draw [r] (rr2) to (rrf0);
      \draw [r] (rrf0) to (rr1);
      \draw [r] (rr1) to (rrf2);      
      \draw [r] (rrf2) to [bend left=22] (rr0);
      %%%%%%%%%%%%%%%%%%%%%%%
      \draw [b] (e0) to (r0);
      \draw [b] (r0) to (rr0);
      \draw [b] (rr0) to [bend left=22] (e0);
      %%
      \draw [b] (rrf1) to (rf1);
      \draw [b] (rf1) to (f1);
      \draw [b] (f1) to [bend right=22] (rrf1);
      %%
      \draw [b] (e2) to (r2);
      \draw [b] (r2) to (rr2);
      \draw [b] (rr2) to [bend left=22] (e2);
      %%
      \draw [b] (rrf0) to (rf0);
      \draw [b] (rf0) to (f0);
      \draw [b] (f0) to [bend left=22] (rrf0);
      %%
      \draw [b] (e1) to (r1);
      \draw [b] (r1) to (rr1);
      \draw [b] (rr1) to [bend right=22] (e1);
      %%
      %%
      \draw [b] (rrf2) to (rf2);
      \draw [b] (rf2) to (f2);
      \draw [b] (f2) to [bend left=22] (rrf2);
    \end{scope}
    %%
    \begin{scope}[shift={(7.75,2.5)}]
      \node (1) at (0:3) [v-small] {$1$};
      \node (r) at (40:3) [v-small] {$r$};
      \node (r2) at (80:3) [v-small] {$r^2$};
      \node (r3) at (120:3) [v-small] {$r^3$};
      \node (r4) at (160:3) [v-small] {$r^4$};
      \node (r5) at (200:3) [v-small] {$r^5$};
      \node (r6) at (240:3) [v-small] {$r^6$};
      \node (r7) at (280:3) [v-small] {$r^7$};
      \node (r8) at (320:3) [v-small] {$r^8$};
      %%
      \node (s) at (0:2) [v-small] {$s$};
      \node (rs) at (40:2) [v-small] {$rs$};
      \node (r2s) at (80:2) [v-small] {$r^2\!s$};
      \node (r3s) at (120:2) [v-small] {$r^3\!s$};
      \node (r4s) at (160:2) [v-small] {$r^4\!s$};
      \node (r5s) at (200:2) [v-small] {$r^5\!s$};
      \node (r6s) at (240:2) [v-small] {$r^6\!s$};
      \node (r7s) at (280:2) [v-small] {$r^7\!s$};
      \node (r8s) at (320:2) [v-small] {$r^8\!s$};
      %%
      \node (s2) at (0:1) [v-small] {$s^2$};
      \node (rs2) at (40:1) [v-small] {$rs^2$};
      \node (r2s2) at (80:1) [v-small] {$r^2\!s^2$};
      \node (r3s2) at (120:1) [v-small] {$r^3\!s^2$};
      \node (r4s2) at (160:1) [v-small] {$r^4\!s^2$};
      \node (r5s2) at (200:1) [v-small] {$r^5\!s^2$};
      \node (r6s2) at (240:1) [v-small] {$r^6\!s^2$};
      \node (r7s2) at (280:1) [v-small] {$r^7\!s^2$};
      \node (r8s2) at (320:1) [v-small] {$r^8\!s^2$};
      %%
      \draw [R9] (1) to (r);
      \draw [R9] (r) to (r2);
      \draw [R9] (r2) to (r3);
      \draw [R9] (r3) to (r4);
      \draw [R9] (r4) to (r5);
      \draw [R9] (r5) to (r6);
      \draw [R9] (r6) to (r7);
      \draw [R9] (r7) to (r8);
      \draw [R9] (r8) to (1);
      %%
      \draw [R3] (s) to (r7s);
      \draw [R3] (r7s) to (r5s);
      \draw [R3] (r5s) to (r3s);
      \draw [R3] (r3s) to (rs);
      \draw [R3] (rs) to (r8s);
      \draw [R3] (r8s) to (r6s);
      \draw [R3] (r6s) to (r4s);
      \draw [R3] (r4s) to (r2s);
      \draw [R3] (r2s) to (s);
      %%
      \draw [r] (s2) to (r4s2);
      \draw [r] (r4s2) to (r8s2);
      \draw [r] (r8s2) to (r3s2);
      \draw [r] (r3s2) to (r7s2);
      \draw [r] (r7s2) to (r2s2);
      \draw [r] (r2s2) to (r6s2);
      \draw [r] (r6s2) to (rs2);
      \draw [r] (rs2) to (r5s2);
      \draw [r] (r5s2) 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);
      \draw [b] (r7) to (r7s); \draw [b] (r7s) to (r7s2);
      \draw [b,bend left] (r7s2) to (r7);
      \draw [b] (r8) to (r8s); \draw [b] (r8s) to (r8s2);
      \draw [b,bend left] (r8s2) to (r8);
    \end{scope}
  \end{tikzpicture}
  \]

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

\item If $A,B\leq G$ and $x\in G$, define the {\em $(A,B)$-double
  coset} to be the set
  \[
  AxB:=\big\{axb\mid a\in A,\,b\in B\big\}.
  \]
  \begin{enumerate}
  \item Show that $G$ is the disjoint union of its $(A,B)$-double cosets. 
  \item Show that if $A$ and $B$ are finite, then
    $|AxB|=[x^{-1}Ax:x^{-1}Ax\cap B]\cdot |B|$.
  \item Find a Cayley graph of a group $G$ of order at least $16$, and
    pick two subgroups, $A$ and $B$, neither normal, that have more
    than two $(A,B)$-double cosets. Partition $G$ by the double
    cosets, and highlight these on the Cayley graph by coloring the
    nodes. Also partition $G$ by the left and right cosets of $AB$, if
    this is a subgroup. Please use an example that is different from
    all of your homework collaborators.
  \end{enumerate}
  
  %%-------------------------------------------------------------------
  
\item The \emph{centralizer} of an element $h\in G$ is the set of
  elements that commute with it:
  \[
  C_G(h):=\big\{g\in G\mid gh=hg\big\}.
  \]
  \begin{enumerate}
    \item Find the centralizers of the elements $r$, $s$, and $r^3$ in both
      groups from Problem~1.
    \item Show that $C_G(H)\normaleq N_G(H)$.
    \item Let $h\in G$ with $[G:\<h\>]=n<\infty$. Show that there are
      exactly $[G:C_G(h)]$ elements conjugate to $h$, by constructing
      a map from the right cosets of $C_G(h)$ to the conjugacy class
      $\cl_G(h)$ of $h$, and showing that it is a well-defined
      bijection.
    \item Partition both groups from Part~(a) by conjugacy classes. 
  \end{enumerate}

  %%-------------------------------------------------------------------
    
\item Let $G$ be a group, not necessarily finite, and $H\leq G$ a
  subgroup.
  \begin{enumerate}
  \item Show that the subgroup
    $
    N:=\bigcap_{x\in G} xHx^{-1}
    $
    is normal in $G$.
  \item Show that every normal subgroup $K\normaleq G$ contained
    in $H$ is contained in $N$. In other words, $N$ is the \emph{largest
      normal subgroup} of $G$ contained in $H$.
  \item Show that if $[G:N]<\infty$ and $|H|=\infty$, then $|H\cap N|>1$.
  \end{enumerate}

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

\item Answer the following questions about the symmetric and
  alternating groups. The subgroup lattice of $G=S_4$ is shown below.
  %%
  \newcommand\Hg{6.5} % height of A_4 
  \newcommand\Hf{5.25} % height of A_4 
  \newcommand\He{4.5} % height of D_4
  \newcommand\Hd{3.75} % height of S_3 
  \newcommand\Hc{3} % height of C_4, V_4 
  \newcommand\Hb{2} % height of C_3
  \newcommand\Ha{1.5} % height of C_2 
  \[
  \begin{tikzpicture}[scale=1.2, shorten >= -2pt, shorten <= -2pt]
    \begin{scope}[scale=1,shift={(0,0)}]
      \tikzstyle{every node}=[font=\normalsize]
      %\tikzstyle{every edge}=[faded]
      \node (S4) at (0,\Hg) {$S_4$};
      \node (A4) at (0,\Hf) {$A_4$};
      \node (D4-1) at (-3,\He) {$D_4$};
      \node (D4-2) at (-2.5,\He) {$D_4$};
      \node (D4-3) at (-2,\He) {$D_4$};
      %%
      \node (S3-1) at (2.25,\Hd) {$S_3$};
      \node (S3-2) at (2.75,\Hd) {$S_3$};
      \node (S3-3) at (3.25,\Hd) {$S_3$};
      \node (S3-4) at (3.75,\Hd) {$S_3$};
      %%
      \node (C4-1) at (-4.25,\Hc) {$C_4$};
      \node (C4-2) at (-3.75,\Hc) {$C_4$};
      \node (C4-3) at (-3.25,\Hc) {$C_4$};
      %%
      \node (V4-0) at (-2.5,\Hc) {$V_4$};
      \node (V4-1) at (-1.75,\Hc) {$V_4$};
      \node (V4-2) at (-1.25,\Hc) {$V_4$};
      \node (V4-3) at (-.75,\Hc) {$V_4$};
      %%
      \node (C3-1) at (2.25,\Hb) {$C_3$};
      \node (C3-2) at (2.75,\Hb) {$C_3$};
      \node (C3-3) at (3.25,\Hb) {$C_3$};
      \node (C3-4) at (3.75,\Hb) {$C_3$};
      %%
      \node (C2-1) at (-3,\Ha) {$C_2$};
      \node (C2-2) at (-2.5,\Ha) {$C_2$};
      \node (C2-3) at (-2,\Ha) {$C_2$};
      %%
      \node (C2-4) at (-1.25,\Ha) {$C_2$};
      \node (C2-5) at (-.75,\Ha) {$C_2$};
      \node (C2-6) at (-.25,\Ha) {$C_2$};
      \node (C2-7) at (.25,\Ha) {$C_2$};
      \node (C2-8) at (.75,\Ha) {$C_2$};
      \node (C2-9) at (1.25,\Ha) {$C_2$};
      \node (1) at (0,0) {$C_1$};
      %%
      \draw[faded][faded] (S4) to (A4);
      \draw[faded] (S4) to (D4-1); \draw[faded] (S4) to (D4-2); \draw[faded] (S4) to (D4-3);
      \draw[faded] (S4) to (S3-1); \draw[faded] (S4) to (S3-2);
      \draw[faded] (S4) to (S3-3); \draw[faded] (S4) to (S3-4);
      %%
      \draw[faded] (D4-1) to (V4-0); \draw[faded] (D4-2) to (V4-0); \draw[faded] (D4-3) to (V4-0);
      \draw[faded] (D4-1) to (C4-1); \draw[faded] (D4-2) to (C4-2); \draw[faded] (D4-3) to (C4-3);
      \draw[faded] (D4-1) to (V4-1); \draw[faded] (D4-2) to (V4-2); \draw[faded] (D4-3) to (V4-3);
      %%
      \draw[faded] (C4-1) to (C2-1); \draw[faded] (C4-2) to (C2-2); \draw[faded] (C4-3) to (C2-3);
      %%
      \draw[faded] (V4-0) to (C2-1); \draw[faded] (V4-0) to (C2-2); \draw[faded] (V4-0) to (C2-3);
      \draw[faded] (V4-1) to (C2-1); \draw[faded] (V4-2) to (C2-2); \draw[faded] (V4-3) to (C2-3);
      \draw[faded] (V4-1) to (C2-4); \draw[faded] (V4-2) to (C2-6); \draw[faded] (V4-3) to (C2-8);
      \draw[faded] (V4-1) to (C2-5); \draw[faded] (V4-2) to (C2-7); \draw[faded] (V4-3) to (C2-9);
      %%
      \draw[faded] (C2-1) to (1); \draw[faded] (C2-2) to (1); \draw[faded] (C2-3) to (1);
      \draw[faded] (C2-4) to (1); \draw[faded] (C2-5) to (1); \draw[faded] (C2-6) to (1);
      \draw[faded] (C2-7) to (1); \draw[faded] (C2-8) to (1); \draw[faded] (C2-9) to (1);
      %%
      \draw[faded] (A4) to (V4-0); \draw[faded] (A4) to (C3-1); \draw[faded] (A4) to (C3-2);
      \draw[faded] (A4) to (C3-3); \draw[faded] (A4) to (C3-4);
      %%
      \draw[faded] (S3-1) to (C3-1); \draw[faded] (S3-2) to (C3-2);
      \draw[faded] (S3-3) to (C3-3); \draw[faded] (S3-4) to (C3-4);
      %%
      \draw[faded] (C3-1) to (1); \draw[faded] (C3-2) to (1);
      \draw[faded] (C3-3) to (1); \draw[faded] (C3-4) to (1);
      %%
      \draw[faded] (S3-1) to (C2-4); \draw[faded] (S3-1) to (C2-5); \draw[faded] (S3-1) to (C2-6);
      \draw[faded] (S3-2) to (C2-4); \draw[faded] (S3-2) to (C2-7); \draw[faded] (S3-2) to (C2-8);
      \draw[faded] (S3-3) to (C2-6); \draw[faded] (S3-3) to (C2-7); \draw[faded] (S3-3) to (C2-9);
      \draw[faded] (S3-4) to (C2-5); \draw[faded] (S3-4) to (C2-8); \draw[faded] (S3-4) to (C2-9);
      %%
      \node[anchor=east] at (-5,\Hg) {\normalsize \textbf{Order} $=24$};
      \node[anchor=east] at (-5,\Hf) {\normalsize $12$};
      \node[anchor=east]  at (-5,\He) {\normalsize $8$};
      \node[anchor=east]  at (-5,\Hd) {\normalsize $6$};
      \node[anchor=east]  at (-5,\Hc) {\normalsize $4$};
      \node[anchor=east]  at (-5,\Hb) {\normalsize $3$};
      \node[anchor=east]  at (-5,\Ha) {\normalsize $2$};
      \node[anchor=east]  at (-5,0) {\normalsize $1$};
      %%
      \node[anchor=east] at (5,\Hg) {\normalsize \textbf{Index} $=1$};
      \node[anchor=east]  at (5,\Hf) {\normalsize $2$};
      \node[anchor=east]  at (5,\He) {\normalsize $3$};
      \node[anchor=east]  at (5,\Hd) {\normalsize $4$};
      \node[anchor=east]  at (5,\Hc) {\normalsize $6$};
      \node[anchor=east]  at (5,\Hb) {\normalsize $8$};
      \node[anchor=east]  at (5,\Ha) {\normalsize $12$};
      \node[anchor=east]  at (5,0) {\normalsize $24$};
    \end{scope}
  \end{tikzpicture}
  \]

  \begin{enumerate}
  \item Partition the subgroups of $S_4$ into conjugacy classes, and justify
    your answer. You can assume that two elements are conjugate iff
    they have the same cycle type.
  \item For each $\cl_{S_4}(H)$, find the isomorphism
    type of the normalizer, $N_{S_4}(H)$.
  \item Compute the centralizers of $e$, $(12)$, $(123)$, $(1234)$,
    and $(12)(34)$ in $S_4$.
  \item Partition the elements of $A_4$ by conjugacy class. Then
    pick one element $\sigma$ from each class, and find its
    centralizer, $C_{A_4}(\sigma)$.
  \item For each of the following elements $\sigma\in S_5$, find
    $|\cl_{S_5}(\sigma)|$, and then its centralizer
    $C_{S_5}(\sigma)$: $e$, $(12)$, $(123)$, $(1234)$, $(12345)$,
    $(12)(34)$, and $(123)(45)$ in $S_5$.
  \end{enumerate}


  
\end{enumerate}

\end{document}

