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\pagestyle{fancy} \lhead{{\sf Homework 7 $|$ Due October 14 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 8510 $|$ Abstract Algebra $|$
    Fall 2022 $|$ M.~Macauley}}

\begin{document}

$\;$ \vspace{-6mm}

\begin{enumerate}

\item The \emph{chutes and ladders diagram} of a finite group $G$ is
  constructed by taking its subgroup lattice, and adding:
  \begin{itemize}
  \item a red arrow for each ``maximal central descent''
    $N\Alert{\searrow}L$, where $L=[G,N]$,
  \item a blue arrow for each ``maximal central ascent'', $N\Balert{\nearrow} Z$, where $Z/N=Z(G/N)$.
  \end{itemize}
  An example is shown below for the semiabelian group $\SA_8=\big\<r,s\mid r^8=s^2=1,\,srs=r^5\big\>$. \vspace{-3mm}
  \[
  \begin{tikzpicture}[scale=1.1]
    \tikzset{every loop/.style={min distance=5mm,looseness=10}}
    \tikzstyle{every node}=[font=\normalsize]
%% Semiabelian group for $n=4$ and $n=5%. 
\tikzstyle{v-small} = [circle, draw, fill=lightgrey,inner sep=0pt, 
  minimum size=6mm]
\tikzstyle{r-bend} = [draw, very thick, Red,-stealth,bend right=15]
\begin{scope}[shift={(-5.75,3.2)},scale=1.4]
  \tikzstyle{every node}=[font=\footnotesize]
      \node (s) at (0:1.1) [v-small] {$s$};
      \node (rs) at (45:1.15) [v-small] {$rs$};
      \node (r2s) at (90:1.15) [v-small] {$r^2\!s$};
      \node (r3s) at (135:1.15) [v-small] {$r^3\!s$};
      \node (r4s) at (180:1.15) [v-small] {$r^4\!s$};
      \node (r5s) at (225:1.15) [v-small] {$r^5\!s$};
      \node (r6s) at (270:1.15) [v-small] {$r^6\!s$};
      \node (r7s) at (315:1.15) [v-small] {$r^7\!s$};
      %%
      \node (1) at (0:2) [v-small] {$1$};
      \node (r) at (45:2) [v-small] {$r$};
      \node (r2) at (90:2) [v-small] {$r^2$};
      \node (r3) at (135:2) [v-small] {$r^3$};
      \node (r4) at (180:2) [v-small] {$r^4$};
      \node (r5) at (225:2) [v-small] {$r^5$};
      \node (r6) at (270:2) [v-small] {$r^6$};
      \node (r7) at (315:2) [v-small] {$r^7$};
      %%
      \draw [r-bend] (1) to (r);
      \draw [r-bend] (r) to (r2);
      \draw [r-bend] (r2) to (r3);
      \draw [r-bend] (r3) to (r4);
      \draw [r-bend] (r4) to (r5);
      \draw [r-bend] (r5) to (r6);
      \draw [r-bend] (r6) to (r7);
      \draw [r-bend] (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) {\large $\SA_8$};
\end{scope}
%%
    \begin{scope}[shift={(0,0)},shorten >= -2pt, shorten <= -2pt]
      \node (G) at (3.5,6) {$\SA_8$};
      \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[faded] (1) -- (s);
      \draw[faded] (1) -- (r4s);
      \draw[faded] (1) -- (r4);
      \draw[faded] (s) -- (r4-s);
      \draw[faded] (r4s) -- (r4-s);
      \draw[faded] (r4) -- (r4-s);
      \draw[faded] (r4) -- (r2s);
      \draw[faded] (r4) -- (r2);
      \draw[faded] (r4-s) -- (r2-s);
      \draw[faded] (r2s) -- (r2-s);
      \draw[faded] (r2) -- (r2-s);
      \draw[faded] (r2) -- (r);
      \draw[faded] (r2) -- (rs);
      \draw[faded] (r2-s) -- (G);
      \draw[faded] (r) -- (G);
      \draw[faded] (rs) -- (G);
      \draw [r] (G) to[bend right=5] (r4);
      \draw [r] (r) to[bend left=30] (r4);
      \draw [r] (rs) to[bend left=20] (r4);
      \draw [r] (r2-s) to[bend right=25] (r4);
      \draw [r] (r2) to [bend left=20] (1);
      \draw [r] (r4-s) to (r4);
      \draw [r] (r2s) to (r4);
      \draw [r] (r4) to [bend right=0] (1);
      \draw [b] (1) to [bend right=45] (r2);
      \draw [b] (r4) to [bend right=10]  (G);
      \draw [b] (r4-s) to [bend left=35] (G);
      \draw [b] (r2s) to [bend left=13] (G);
      \draw [b] (r) to [bend right=30] (G);
      \draw [b] (rs) to (G);
      \draw [b] (r2-s) to [bend left=15] (G);
      \draw [b] (r2) to [bend right=30] (G);
      \path (G) edge [b,loop above,-stealth] (G);
      \path (1) edge [r,loop left,-stealth] (1);
    \end{scope}
  \end{tikzpicture}
  \]
\begin{enumerate}
\item Construct the chutes and ladders diagram of the following
  groups: $C_4\rtimes C_4$, $\DQ_8$, $\SA_{32}$, $Q_{32}$, and $Q_{64}$.
  \[
  \hspace*{-12mm}
  \begin{tikzpicture}[shorten >= -2pt, shorten <= -2pt,scale=1.2,yscale=1.35]
    \tikzstyle{every node}=[font=\footnotesize]
    \begin{scope}[shift={(0,0)},xscale=.95]
      \node (G) at (0,4) {$C_4\rtimes C_4$};
      \node (C4xC2-1) at (-1.5,3) {$C_4\times C_2$};
      \node (C4xC2-2) at (0,3) {$C_4\times C_2$};
      \node (C4xC2-3) at (1.5,3) {$C_4\times C_2$};
      \node (C4-1) at (-3,2) {$C_4$};
      \node (C4-2) at (-2,2) {$C_4$};
      \node (C4-3) at (-1.33,2) {$C_4$};
      \node (C4-4) at (-.67,2) {$C_4$};
      \node (C4-5) at (1.5,2) {$C_4$};
      \node (C4-6) at (3,2) {$C_4$};
      \node (V4) at (0,2) {$V_4$};
      \node (C2-1) at (-1.5,1) {$C_2$};
      \node (C2-2) at (0,1) {$C_2$};
      \node (C2-3) at (1.5,1) {$C_2$};
      \node (1) at (0,0) {$\<1\>$};
      \draw[faded] (G) to (C4xC2-1); \draw[faded] (G) to (C4xC2-2); \draw[faded] (G) to (C4xC2-3);
      \draw[faded] (C4xC2-1) to (C4-1);
      \draw[faded] (C4xC2-1) to (C4-2);
      \draw[faded] (C4xC2-1) to (V4);
      \draw[faded] (C4xC2-2) to (C4-3);
      \draw[faded] (C4xC2-2) to (C4-4);
      \draw[faded] (C4xC2-2) to (V4); 
      \draw[faded] (C4xC2-3) to (C4-5);
      \draw[faded] (C4xC2-3) to (C4-6);
      \draw[faded] (C4xC2-3) to (V4);
      \draw[faded] (C4-1) to (C2-1); \draw[faded] (C4-2) to (C2-1);
      \draw[faded] (C4-3) to (C2-1); \draw[faded] (C4-4) to (C2-1);
      \draw[faded] (V4) to (C2-1); \draw[faded] (V4) to (C2-2); \draw[faded] (V4) to (C2-3);
      \draw[faded] (C4-5) to (C2-3); \draw[faded] (C4-6) to (C2-3);
      \draw[faded] (C2-1) to (1); \draw[faded] (C2-2) to (1); \draw[faded] (C2-3) to (1);
      \end{scope}
    %%
    \begin{scope}[shift={(7,1)}]
    \node (G) at (0,3) {$\DQ_8$};
    %%
    \node (C4xC2-1) at (-3,2) {$C_4\!\times\!C_2$}; % 010,001,011
    \node (D4-1) at (-2,2) {$D_4$}; % 100,001,101    
    \node (C4xC2-2) at (-1,2) {$C_4\!\times\!C_2$}; % 100,010,110
    \node (D4-2) at (0,2) {$D_4$};  % 100,011,111
    \node (C4xC2-3) at (1,2) {$C_4\!\times\!C_2$};  % 010,101,111
    \node (D4-3) at (2,2) {$D_4$};  % 110,001,111
    \node (Q8) at (3,2) {$Q_8$};  % 110,011,101
    %%
    \node (V4-12) at (-3,1) {$V_4$};
    \node (V4-13) at (-2,1) {$V_4$};
    \node (V4-23) at (-1,1) {$V_4$};
    \node (Z) at (0,1) {$C_4$};
    \node (C4-1) at (1,1) {$C_4$};
    \node (C4-2) at (2,1) {$C_4$};
    \node (C4-3) at (3,1) {$C_4$};
    %%
    \node (C2-11) at (-3.25,0) {$C_2$};
    \node (C2-12) at (-2.75,0) {$C_2$};
    \node (C2-21) at (-2.25,0) {$C_2$};
    \node (C2-22) at (-1.75,0) {$C_2$};
    \node (C2-31) at (-1.25,0) {$C_2$};
    \node (C2-32) at (-0.75,0) {$C_2$};    
    \node (-I) at (1,0) {$C_2$};
    %%
    \node (I) at (0,-1) {$C_1$};
    %%
    %%
    \draw[faded] (C4xC2-1) to (G); \draw[faded] (D4-1) to (G); \draw[faded] (C4xC2-2) to (G);
    \draw[faded] (D4-2) to (G); \draw[faded] (C4xC2-3) to (G); \draw[faded] (D4-3) to (G);
    \draw[faded] (Q8) to (G);
    %%
    \draw[faded] (C4xC2-1) to (Z); \draw[faded] (C4xC2-2) to (Z); \draw[faded] (C4xC2-3) to (Z);
    \draw[faded] (V4-12) to (D4-1); \draw[faded] (V4-12) to (D4-2); \draw[faded] (V4-12) to (C4xC2-3);
    \draw[faded] (V4-13) to (D4-1); \draw[faded] (V4-13) to (D4-3); \draw[faded] (V4-13) to (C4xC2-2);
    \draw[faded] (V4-23) to (D4-2); \draw[faded] (V4-23) to (D4-3); \draw[faded] (V4-23) to (C4xC2-1);
    %%
    \draw[faded] (C4-1) to (C4xC2-1); \draw[faded] (C4-1) to (D4-1);
    \draw[faded] (C4-2) to (C4xC2-2); \draw[faded] (C4-2) to (D4-2);
    \draw[faded] (C4-3) to (C4xC2-3); \draw[faded] (C4-3) to (D4-3);
    %%
    \draw[faded] (Q8) to (C4-1); \draw[faded] (Q8) to (C4-2); \draw[faded] (Q8) to (C4-3);
    %%
    \draw[faded] (V4-12) to (C2-11); \draw[faded] (V4-12) to (C2-12); 
    \draw[faded] (V4-13) to (C2-21); \draw[faded] (V4-13) to (C2-22); 
    \draw[faded] (V4-23) to (C2-31); \draw[faded] (V4-23) to (C2-32); 
    %%
    \draw[faded] (-I) to (V4-12); \draw[faded] (-I) to (V4-13); \draw[faded] (-I) to (V4-23);
    \draw[faded] (-I) to (C4-1); \draw[faded] (-I) to (C4-2); \draw[faded] (-I) to (C4-3);
    \draw[faded] (-I) to (Z);
    %%
    \draw[faded] (C2-11) to (I); \draw[faded] (C2-12) to (I); 
    \draw[faded] (C2-21) to (I); \draw[faded] (C2-22) to (I); 
    \draw[faded] (C2-31) to (I); \draw[faded] (C2-32) to (I);
    \draw[faded] (-I) to (I);
    %%
  \end{scope}
    \end{tikzpicture}
    \]
  
\item On the subgroups lattices of the groups from Part~(a), mark the
  upper and lower central series, the derived series, and determine
  the abelianization.
\item Two functions $f\colon X\to Y$ and $g\colon Y\to X$ are
  \emph{generalized inverses} if $f\circ g\circ f=f$ and $g\circ
  f\circ g=g$. Show that maximal central ascents and maximal central
  descents are generalized inverses on the set of normal subgroups of
  a groups $G$.
\end{enumerate}

\[
\hspace*{-8mm}
\begin{tikzpicture}[node distance=1cm,shorten >= -2pt, shorten <= -2pt,
      scale=.9]
    \begin{scope}[shift={(1,0)},shift={(0,1.5)},xscale=.94]
      \tikzstyle{every node}=[font=\small]
      \node(G) at (3.5,6) {$\SA_{32}$};
      \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(r8-s) at (-1.75,1.5) {$\<r^8,s\>$};
      \node(r4s) at (0,1.5) {$\<r^4s\>$};
      \node(r4) at (1.75,1.5) {$\<r^4\>$};
      \node(s) at (-3.5,0) {$\<s\>$};      
      \node(r8s) at (-1.75,0) {$\<r^8s\>$};
      \node(r8) at (0,0) {$\<r^8\>$};
      \node(1) at (-1.75,-1.5) {$\<1\>$};
      \draw[faded] (1) to (s);
      \draw[faded] (1) to (r8s);
      \draw[faded] (1) to (r8);
      \draw[faded] (s) to (r8-s);
      \draw[faded] (r8s) to (r8-s);
      \draw[faded] (r8) to (r8-s);
      \draw[faded] (r8) to (r4s);
      \draw[faded] (r8) to (r4);
      \draw[faded] (r8-s) to (r4-s);
      \draw[faded] (r4s) to (r4-s);
      \draw[faded] (r4) to (r4-s);
      \draw[faded] (r4) to (r2s);
      \draw[faded] (r4) to (r2);
      \draw[faded] (r4-s) to (r2-s);
      \draw[faded] (r2s) to (r2-s);
      \draw[faded] (r2) to (r2-s);
      \draw[faded] (r2) to (r);
      \draw[faded] (r2) to (rs);
      \draw[faded] (r2-s) to (G);
      \draw[faded] (r) to (G);
      \draw[faded] (rs) to (G);
    \end{scope}
     \begin{scope}[shift={(9,0)},scale=1.5]
     \node (Q32) at (0,5) {$Q_{32}$};
     \node (C16) at (0,4) {$C_{16}$};
     \node (C8) at (0,3) {$C_8$};
     \node (C4) at (0,2) {$C_4$};
     \node (C2) at (0,1) {$C_2$};
     \node (1) at (0,0) {$C_1$};
     \node (Q16-l) at (-1,4) {$Q_{16}$};
     \node (Q8-l) at (-1,3) {${}_{\color{midgrey} 2}Q_8$};
     \node (C4-l) at (-1,2) {${}_{\color{midgrey} 4}C_4$};
     \node (Q16-r) at (1,4) {$Q_{16}$};
     \node (Q8-r) at (1,3) {${}_{\color{midgrey} 2}Q_8$};
     \node (C4-r) at (1,2) {${}_{\color{midgrey} 4}C_4$};
     \draw[faded] (Q32) to (C16);
     \draw[faded] (C16) to (C8);
     \draw[faded] (C8) to (C4);
     \draw[faded] (C4) to (C2);
     \draw[faded] (C2) to (1);
     \draw[faded] (Q32) to (Q16-l); \draw[faded] (Q32) to (Q16-r); 
     \draw[faded] (Q16-l) to (Q8-l); \draw[faded] (Q16-r) to (Q8-r); 
     \draw[faded] (Q8-l) to (C4-l); \draw[faded] (Q8-r) to (C4-r); 
     \draw[faded] (Q16-l) to (C8); \draw[faded] (Q16-r) to (C8);
     \draw[faded] (Q8-l) to (C4); \draw[faded] (Q8-r) to (C4);
     \draw[faded] (C4-l) to (C2); \draw[faded] (C4-r) to (C2);
  \end{scope}
  %%
  \begin{scope}[shift={(14,0)},scale=1.5]
     \node (Q64) at (0,6) {$Q_{64}$};
     \node (C32) at (0,5) {$C_{32}$};
     \node (C16) at (0,4) {$C_{16}$};
     \node (C8) at (0,3) {$C_8$};
     \node (C4) at (0,2) {$C_4$};
     \node (C2) at (0,1) {$C_2$};
     \node (1) at (0,0) {$C_1$};
     \node (Q32-l) at (-1,5) {$Q_{32}$};
     \node (Q16-l) at (-1,4) {${}_{\color{midgrey} 2}Q_{16}$};
     \node (Q8-l) at (-1,3) {${}_{\color{midgrey} 4}Q_8$};
     \node (C4-l) at (-1,2) {${}_{\color{midgrey} 8}C_4$};
     \node (Q32-r) at (1,5) {$Q_{32}$};
     \node (Q16-r) at (1,4) {${}_{\color{midgrey} 2}Q_{16}$};
     \node (Q8-r) at (1,3) {${}_{\color{midgrey} 4}Q_8$};
     \node (C4-r) at (1,2) {${}_{\color{midgrey} 8}C_4$};
     \draw[faded] (Q64) to (C32);
     \draw[faded] (C32) to (C16);
     \draw[faded] (C16) to (C8);
     \draw[faded] (C8) to (C4);
     \draw[faded] (C4) to (C2);
     \draw[faded] (C2) to (1);
     \draw[faded] (Q64) to (Q32-l); \draw[faded] (Q64) to (Q32-r);
     \draw[faded] (Q32-l) to (Q16-l); \draw[faded] (Q32-r) to (Q16-r); 
     \draw[faded] (Q16-l) to (Q8-l); \draw[faded] (Q16-r) to (Q8-r); 
     \draw[faded] (Q8-l) to (C4-l); \draw[faded] (Q8-r) to (C4-r); 
     \draw[faded] (Q32-l) to (C16); \draw[faded] (Q32-r) to (C16);
     \draw[faded] (Q16-l) to (C8); \draw[faded] (Q16-r) to (C8);
     \draw[faded] (Q8-l) to (C4); \draw[faded] (Q8-r) to (C4);
     \draw[faded] (C4-l) to (C2); \draw[faded] (C4-r) to (C2);
  \end{scope}
  \end{tikzpicture}
  \]



\item Let $G$ be a group with nilpotency class $n<\infty$.
  \begin{enumerate}
  \item Show that every subgroup of $G$ has nilpotency class at most $n$.
  \item Show that every homomorphic image of $G$ has nilpotency class
    at most $n$.
  \item Show that nontrivial normal subgroups of $G$ intersect the
    center nontrivially. \\
  \end{enumerate}


  
\item Let $G$ be a finite group whose inner automorphism group
  $\Inn(G)$ is abelian. Show that $G'\leq Z(G)$, and use this to
  conclude that $G$ is nilpotent. \\



  
\item Let $G$ be a finite group in which every maximal subgroup is normal.
  \begin{enumerate}
  \item Prove that $G$ is nilpotent. [\emph{Hint}: If not, then take a
    non-normal Sylow subgroup $P\leq G$, and choose a maximal $M\leq
    G$ containing $N_G(P)$.]
  \item Show that every maximal subgroup of $G$ has prime index. \\
  \end{enumerate}


  
\item Show that if a finite group $G$ has no fully unnormal subgroups,
  then it is nilpotent. [\emph{Hint}: Recall from HW 5 that if $P$ is
    a Sylow $p$-subgroup of $H\normaleq G$, then $G=N_G(P)H$.] \\


  
\end{enumerate}

\end{document}
