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\pagestyle{fancy} \lhead{{\sf Homework 12 $|$ Due November 12 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 4120/6120 $|$ Modern Algebra $|$
    Fall 2021 $|$ M.~Macauley}}

\begin{document}

%$\;$

\begin{enumerate}

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

\item Let $G$ be an unknown group of order $8$. If it has no element of order $4$, then $g^2=e$ for all $g\in G$, and so $G$ must be abelian. Otherwise, it
      has a ``partial Cayley diagram'' like one of the following: 
    \colorlet{lightgrey}{black!15}
    \tikzstyle{v} = [circle, draw, fill=lightgrey,inner sep=0pt, 
      minimum width=6mm]
     \[
    \begin{tikzpicture}[scale=1.5,auto]
      \tikzstyle{every node}=[font=\footnotesize]
      \node (b) at (0,0) [v] {$b$};
      \node (e) at (0,1) [v] {$1$};
      \node (ba) at (1,0) [v] {$ba$};
      \node (a) at (1,1) [v] {$a$};
      \node (ba2) at (2,0) [v] {$ba^{\!2}$};
      \node (a2) at (2,1) [v] {$a^{\!2}$};
      \node (ba3) at (3,0) [v] {$ba^{\!3}$};
      \node (a3) at (3,1) [v] {$a^{\!3}$};
      \draw [r] (e) to (a);
      \draw [r] (a) to (a2);
      \draw [r] (a2) to (a3);
      \draw [r] (a3) to [bend right=25] (e);
      \draw [r] (b) to (ba);
      \draw [r] (ba) to (ba2);
      \draw [r] (ba2) to (ba3);
      \draw [r] (ba3) to [bend left=25] (b);
      \draw [bb] (e) to (b);
      %%
      \begin{scope}[shift={(5,0)}]
        \node (b) at (0,0) [v] {$b$};
        \node (e) at (0,1) [v] {$1$};
        \node (ba) at (1,0) [v] {$ba$};
        \node (a) at (1,1) [v] {$a$};
        \node (ba2) at (2,0) [v] {$ba^{\!2}$};
        \node (a2) at (2,1) [v] {$a^{\!2}$};
        \node (ba3) at (3,0) [v] {$ba^{\!3}$};
        \node (a3) at (3,1) [v] {$a^{\!3}$};
        \draw [r] (e) to (a);
        \draw [r] (a) to (a2);
        \draw [r] (a2) to (a3);
        \draw [r] (a3) to [bend right=25] (e);
        \draw [r] (b) to (ba);
        \draw [r] (ba) to (ba2);
        \draw [r] (ba2) to (ba3);
        \draw [r] (ba3) to [bend left=25] (b);
        \draw [b] (e) to (b);
      \end{scope}
    \end{tikzpicture}
    \]
    Find all possibilities for finishing each diagram, and label by isomorphism type. \bigskip
  %\end{enumerate}

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

\item The subgroup lattice of the symmetric group $S_4$ is shown
  below. \vspace{-3mm}
  
  \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=.95, shorten >= -2pt, shorten <= -2pt]
    \begin{scope}[scale=1.45,shift={(0,0)}]
      \tikzstyle{every node}=[font=\small]
      \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 (S4) to (A4);
      \draw (S4) to (D4-1); \draw (S4) to (D4-2); \draw (S4) to (D4-3);
      \draw (S4) to (S3-1); \draw (S4) to (S3-2);
      \draw (S4) to (S3-3); \draw (S4) to (S3-4);
      %%
      \draw (D4-1) to (V4-0); \draw (D4-2) to (V4-0); \draw (D4-3) to (V4-0);
      \draw (D4-1) to (C4-1); \draw (D4-2) to (C4-2); \draw (D4-3) to (C4-3);
      \draw (D4-1) to (V4-1); \draw (D4-2) to (V4-2); \draw (D4-3) to (V4-3);
      %%
      \draw (C4-1) to (C2-1); \draw (C4-2) to (C2-2); \draw (C4-3) to (C2-3);
      %%
      \draw (V4-0) to (C2-1); \draw (V4-0) to (C2-2); \draw (V4-0) to (C2-3);
      \draw (V4-1) to (C2-1); \draw (V4-2) to (C2-2); \draw (V4-3) to (C2-3);
      \draw (V4-1) to (C2-4); \draw (V4-2) to (C2-6); \draw (V4-3) to (C2-8);
      \draw (V4-1) to (C2-5); \draw (V4-2) to (C2-7); \draw (V4-3) to (C2-9);
      %%
      \draw (C2-1) to (1); \draw (C2-2) to (1); \draw (C2-3) to (1);
      \draw (C2-4) to (1); \draw (C2-5) to (1); \draw (C2-6) to (1);
      \draw (C2-7) to (1); \draw (C2-8) to (1); \draw (C2-9) to (1);
      %%
      \draw (A4) to (V4-0); \draw (A4) to (C3-1); \draw (A4) to (C3-2);
      \draw (A4) to (C3-3); \draw (A4) to (C3-4);
      %%
      \draw (S3-1) to (C3-1); \draw (S3-2) to (C3-2);
      \draw (S3-3) to (C3-3); \draw (S3-4) to (C3-4);
      %%
      \draw (C3-1) to (1); \draw (C3-2) to (1);
      \draw (C3-3) to (1); \draw (C3-4) to (1);
      %%
      \draw (S3-1) to (C2-4); \draw (S3-1) to (C2-5); \draw (S3-1) to (C2-6);
      \draw (S3-2) to (C2-4); \draw (S3-2) to (C2-7); \draw (S3-2) to (C2-8);
      \draw (S3-3) to (C2-6); \draw (S3-3) to (C2-7); \draw (S3-3) to (C2-9);
      \draw (S3-4) to (C2-5); \draw (S3-4) to (C2-8); \draw (S3-4) to (C2-9);
    \end{scope}
  \end{tikzpicture}
  \]
  \begin{enumerate}
  \item Partition the subgroups into conjugacy classes. Carefully and
    completely justify your answers using the Sylow theorems, without
    making reference to cycle type.
  \item For each conjugacy class $\cl_G(H)$, find the isomorphism type of the
    normalizer $N_G(H)$.
  \item Using the GroupNames website, make a table of all 15 groups of
    order $24$, the number of subgroups, and basic information
    about their Sylow $p$-subgroups (number and isomorphism
    type). Write down at least one observation that you find
    interesting.
  \item Which groups are \emph{not} an internal
    direct or semidirect product of Sylow subgroups?
  \item None of the following groups are among the 15 listed on
    GroupNames: $D_6\times C_2$, $C_6\rtimes C_4$, $C_6\rtimes C_2^2$,
    $C_4\rtimes C_6$, $C_3\rtimes C_2^3$, $C_2^3\rtimes C_3$,
    $C_2^2\rtimes C_6$, $C_3\rtimes Q_8$, $Q_8\rtimes C_3$,
    $C_4\rtimes S_3$. Find which of the 15 each is isomorphic to, and add
    this this to your table.
  \end{enumerate}

  \newpage
  
  %%--------------------------------------------------------------------
  
\item Show that there are no simple groups of the following order. 
  \begin{multicols}{4}
  \begin{enumerate}[(i)]
  \item $45=3^2\cdot 5$
  \item $56=2^3\cdot 7$
  \item $108=2^3\cdot 3^2$
  \item $p^n$. 
  \end{enumerate}
  \end{multicols}
  [\emph{Hint}: For Part~(d), first use a suitable group action to
    show that $|Z(G)|>1$.] \bigskip
  
  %%--------------------------------------------------------------------
  
\item After $A_5$, the next smallest nonabelian simple group is
  $G=\GL_3(\Z_2)$, the invertible $3\times 3$ binary matrices. It has
  order $168=2^3\cdot 3\cdot 7$.
  \begin{enumerate}
  \item What do the Sylow theorems tell us about the possibilities for
    $n_2$, $n_3$, and $n_7$?
  \item Show that $G$ is isomorphic to a subgroup of
    $A_8$. [\emph{Hint}: Let $G$ act on its Sylow $7$-subgroups by
    conjugation.] \bigskip
  \end{enumerate}

  %%--------------------------------------------------------------------
  
\item Let $G$ be an unknown group of order $90$.
  \begin{enumerate}
  \item Using the Sylow theorems, find all possibilities for $n_2$,
    $n_3$, and $n_5$, where $n_p$ is the number of Sylow $p$-subgroups
    of $G$.
  \item Suppose that $G$ has a nonnormal Sylow $5$-subgroup. Show
    that there is a non-trivial homomorphism $\phi\colon G\to S_6$.
  \item If $\phi\colon G\to A_6$, show that $\phi$ is not
    injective. You may assume that $A_6$ is simple.
  \item Show that $G$ is not simple. 
  \end{enumerate} 
  
\end{enumerate}

\end{document}



