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\pagestyle{fancy} \lhead{{\sf Homework 6 $|$ Due March 3 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 4130 $|$ Visual Algebra II $|$
    Spring 2023 $|$ M.~Macauley}}

\begin{document}

$\;$ \vspace{-6mm}

\begin{enumerate}

\item The group $G=\SL_2(\Z_3)$ is solvable with derived length $3$. This means that it can be constructed as a sequence of abelian extensions. Moreover, this can be done two ways:
  \begin{itemize}
  \item From ``top-to-bottom'': $G=G_0\unrhd G_1 \unrhd G_2 \unrhd G_3=\<1\>$.
  \item From ``bottom-to-top'': $\<1\>=G_3\normaleq G_2 \normaleq G_1
    \normaleq G_0=G$.
  \end{itemize}
  These extensions can be encoded by short exact sequences, as shown below. 
  \[
  \begin{tikzpicture}[node distance=1cm,shorten >= -2pt, shorten <= -2pt,scale=.9]
    %%
    \begin{scope}[shift={(0,0)},xscale=.8]
    \tikzstyle{every node}=[font=\footnotesize]
      %\node at (-3,2.1) {\scriptsize $\frac{Q_4}{C_2}\!\cong\!V_4$,};
      \node (1) {$\hspace{-6mm}\bm{G'''\!=\!C_1}$};
      \node (a3) at (-2.5,2) {$\hspace{-8mm}\bm{G''\!=\!C_2}$};
      \node (a2) at (.9,3) {$C_3$};
      \node (b2) at (2,3) {$C_3$};
      \node (abab) at (3.1,3) {$C_3$};
      \node (baba) at (4.35,3) {$C_3$};
      \node (a2b) at (-3.75,4) {$C_4$};
      \node (aba) at (-2.5,4) {$C_4$};
      \node (ab2) at (-1.25,4) {$C_4$};
      \node (a) at (.5,5) {$C_6$};
      \node (b) at (1.6,5) {$C_6$};
      \node (ab) at (2.7,5) {$C_6$};
      \node (ba) at (3.95,5) {$C_6$};
      \node (a2b-ab2) at (-2.5,6) {$\hspace{-6mm}\bm{G'\!=\!Q_8}$};
      \node (G) at (0,8.25) {$\hspace{-6mm}\bm{G\!=\!}\SL_2(\Z_3)$};
      \draw[faded] (1) -- (a3);
      \draw[faded] (1) -- (a2);
      \draw[faded] (1) -- (b2);
      \draw[faded] (1) -- (abab);
      \draw[faded] (1) -- (baba);
      \draw[faded] (a3) -- (a2b);
      \draw[faded] (a3) -- (aba);
      \draw[faded] (a3) -- (ab2);
      \draw[faded] (a3) -- (a);
      \draw[faded] (a3) -- (b);
      \draw[faded] (a3) -- (ab);
      \draw[faded] (a3) -- (ba);
      \draw[faded] (a2) -- (a);
      \draw[faded] (b2) -- (b);
      \draw[faded] (abab) -- (ab);
      \draw[faded] (baba) -- (ba);
      \draw[faded] (a2b) -- (a2b-ab2);
      \draw[faded] (aba) -- (a2b-ab2);
      \draw[faded] (ab2) -- (a2b-ab2);
      \draw[faded] (G) -- (a2b-ab2);
      \draw[faded] (G) -- (a);
      \draw[faded] (G) -- (b);
      \draw[faded] (G) -- (ab);
      \draw[faded] (G) -- (ba);
    \end{scope}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{scope}[shift={(7.75,8)},scale=1.6]
    \tikzstyle{every node}=[font=\footnotesize]
    \node (1L-0) at (0,0) {$1$};
    \node (A-0) at (1,0) {$Q_8$};
    \node (B-0) at (2,0) {\scriptsize $\SL_2(\Z_3)$};
    \node (C-0) at (3,0) {$C_3$};
    \node (1R-0) at (4,0) {$1$};
    \draw[->] (1L-0) to (A-0); \draw[right hook->] (A-0) to (B-0);
    \draw[->>] (B-0) to (C-0); \draw[->] (C-0) to (1R-0);
    %%
    \node (1L-1) at (-1,-1) {$1$};
    \node (A-1) at (0,-1) {$C_2$};
    \node (B-1) at (1,-1) {$Q_8$};
    \node (C-1) at (2,-1) {$V_4$};
    \node (1R-1) at (3,-1) {$1$};
    \draw[->] (1L-1) to (A-1); \draw[right hook->] (A-1) to (B-1);
    \draw[->>] (B-1) to (C-1); \draw[->] (C-1) to (1R-1);
    %%
    \node (1L-2) at (-2,-2) {$1$};
    \node (A-2) at (-1,-2) {$1$};
    \node (B-2) at (0,-2) {$C_2$};
    \node (C-2) at (1,-2) {$C_2$};
    \node (1R-2) at (2,-2) {$1$};
    \draw[->] (1L-2) to (A-2); \draw[right hook->] (A-2) to (B-2);
    \draw[->>] (B-2) to (C-2); \draw[->] (C-2) to (1R-2);
    %%
    \draw[->,dashed] (A-0) to (B-1);
    \draw[->,dashed] (A-1) to (B-2);
\end{scope}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{scope}[shift={(7.75,3)},scale=1.6]
    \tikzstyle{every node}=[font=\footnotesize]
    %%
    \node (1L-0) at (0,0) {$1$};
    \node (A-0) at (1,0) {$C_3$};
    \node (B-0) at (2,0) {$C_3$};
    \node (C-0) at (3,0) {$1$};
    \node (1R-0) at (4,0) {$1$};
    \draw[->] (1L-0) to (A-0); \draw[right hook->] (A-0) to (B-0);
    \draw[->>] (B-0) to (C-0); \draw[->] (C-0) to (1R-0);
    %%
    \node (1L-1) at (-1,-1) {$1$};
    \node (A-1) at (0,-1) {$V_4$};
    \node (B-1) at (1,-1) {$A_4$};
    \node (C-1) at (2,-1) {$C_3$};
    \node (1R-1) at (3,-1) {$1$};
    \draw[->] (1L-1) to (A-1); \draw[right hook->] (A-1) to (B-1);
    \draw[->>] (B-1) to (C-1); \draw[->] (C-1) to (1R-1);
    %%
    \node (1L-2) at (-2,-2) {$1$};
    \node (A-2) at (-1,-2) {$C_2$};
    \node (B-2) at (0,-2) {$\SL_2(\Z_3)$};
    \node (C-2) at (1,-2) {$A_4$};
    \node (1R-2) at (2,-2) {$1$};
    \draw[->] (1L-2) to (A-2); \draw[right hook->] (A-2) to (B-2);
    \draw[->>] (B-2) to (C-2); \draw[->] (C-2) to (1R-2);
    %%
    \draw[->,dashed] (C-1) to (B-0);
    \draw[->,dashed] (C-2) to (B-1);
 \end{scope}
%%
  \end{tikzpicture}
\]
Carry out this construction for the symmetric group $G=S_4$, which has derived length $3$. 

  
  
\item For both of the following groups $G$ shown below, find all
  compositions series (up to isomorphism) and the composition factors. Then repeat the previous exercise, and find the abelianization of $G$.
  \[
  \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}[shorten >= -2pt, shorten <= -2pt,scale=1.2]
    %%
    \begin{scope}[shift={(9,0)},xscale=1.3]
    \tikzstyle{every node}=[font=\footnotesize];
      %%
      \node (G) at (-1.2,\Hg) {$D_3\!\times\!C_4$};
      \node (D6) at (-2.2,\Hf) {$D_6$};
      \node (C12) at (-1.2,\Hf) {$C_{12}$};
      \node (Dic6) at (-.2,\Hf) {$\Dic_6$};
      \node (C4xC2-1) at (1.2,\He) {\footnotesize $C_4\!\!\times\!\!C_2$};
      \node (C4xC2-2) at (1.9,\He) {\footnotesize $C_4\!\!\times\!\!C_2$};
      \node (C4xC2-3) at (2.6,\He) {\footnotesize $C_4\!\!\times\!\!C_2$};
      \node (D3-1) at (-3.2,\Hd) {$D_3$};
      \node (D3-2) at (-2.2,\Hd) {$D_3$};
      \node (C6) at (-1.2,\Hd) {$C_6$};
      \node (V4-1) at (0,\Hc) {$V_4$};
      \node (V4-2) at (.3,\Hc) {$V_4$};
      \node (V4-3) at (.6,\Hc) {$V_4$};
      \node (C4-1) at (1.2,\Hc) {$C_4$};
      \node (C4-2-1) at (1.7,\Hc) {$C_4$};
      \node (C4-2-2) at (2,\Hc) {$C_4$};
      \node (C4-2-3) at (2.3,\Hc) {$C_4$};            
      \node (C3) at (-2.2,\Hb) {$C_3$};
      \node (C2-1-1) at (-1.3,\Ha) {$C_2$};
      \node (C2-1-2) at (-1,\Ha) {$C_2$};
      \node (C2-1-3) at (-.7,\Ha) {$C_2$};
      \node (C2-2-1) at (.1,\Ha) {$C_2$};
      \node (C2-2-2) at (.4,\Ha) {$C_2$};
      \node (C2-2-3) at (.7,\Ha) {$C_2$};
      \node (C2-3) at (1.2,\Ha) {$C_2$};
      \node (1) at (0,0) {$C_1$};
      %%
      %%
      \draw[faded] (G) to (D6); \draw[faded] (G) to (C12); 
      \draw[faded] (G) to (Dic6); 
      \draw[faded] (G) to (C4xC2-1); \draw[faded] (G) to (C4xC2-2); \draw[faded] (G) to (C4xC2-3); 
      %%
      \draw[faded] (D6) to (D3-1); \draw[faded] (D6) to (D3-2); 
      \draw[faded] (D6) to (C6); 
      \draw[faded] (D6) to (V4-1);  \draw[faded] (D6) to (V4-2);  \draw[faded] (D6) to (V4-3);
      %%
      \draw[faded] (C12) to (C6); \draw[faded] (C12) to (C4-1);
      %%
      \draw[faded] (Dic6) to (C6); 
      \draw[faded] (Dic6) to (C4-2-1); \draw[faded] (Dic6) to (C4-2-2); \draw[faded] (Dic6) to (C4-2-3);
      %%
      \draw[faded] (C4xC2-1) to (V4-1); \draw[faded] (C4xC2-2) to (V4-2); \draw[faded] (C4xC2-3) to (V4-3); 
      \draw[faded] (C4xC2-1) to (C4-1); \draw[faded] (C4xC2-2) to (C4-1); \draw[faded] (C4xC2-3) to (C4-1);
      \draw[faded] (C4xC2-1) to (C4-2-1); \draw[faded] (C4xC2-2) to (C4-2-2); \draw[faded] (C4xC2-3) to (C4-2-3);
      %%
      \draw[faded] (D3-1) to (C2-1-1); \draw[faded] (D3-1) to (C2-1-2); \draw[faded] (D3-1) to (C2-1-3); 
      \draw[faded] (D3-1) to (C3);
      \draw[faded] (D3-2) to (C2-2-1); \draw[faded] (D3-2) to (C2-2-2); \draw[faded] (D3-2) to (C2-2-3); 
      \draw[faded] (D3-2) to (C3);
      %%
      \draw[faded] (C6) to (C3); \draw[faded] (C6) to (C2-3); 
      %%
      \draw[faded] (C4-1) to (C2-3); 
      \draw[faded] (C4-2-1) to (C2-3); \draw[faded] (C4-2-2) to (C2-3); \draw[faded] (C4-2-3) to (C2-3);
      %%
      \draw[faded] (V4-1) to (C2-1-1); \draw[faded] (V4-1) to (C2-2-1); \draw[faded] (V4-1) to (C2-3); 
      \draw[faded] (V4-2) to (C2-1-2); \draw[faded] (V4-2) to (C2-2-2); \draw[faded] (V4-2) to (C2-3); 
      \draw[faded] (V4-3) to (C2-1-3); \draw[faded] (V4-3) to (C2-2-3); \draw[faded] (V4-3) to (C2-3);       
      %%
      \draw[faded] (C2-1-1) to (1); \draw[faded] (C2-1-2) to (1); \draw[faded] (C2-1-3) to (1);
      \draw[faded] (C2-2-1) to (1); \draw[faded] (C2-2-2) to (1); \draw[faded] (C2-2-3) to (1);
      \draw[faded] (C2-3) to (1); \draw[faded] (C3) to (1);
      %%
      \node[anchor=east] at (-4,\Hg) {\small \textbf{Order} $=24$};
      \node[anchor=east] at (-4,\Hf) {\small $12$};
      \node[anchor=east]  at (-4,\He) {\small $8$};
      \node[anchor=east]  at (-4,\Hd) {\small $6$};
      \node[anchor=east]  at (-4,\Hc) {\small $4$};
      \node[anchor=east]  at (-4,\Hb) {\small $3$};
      \node[anchor=east]  at (-4,\Ha) {\small $2$};
      \node[anchor=east]  at (-4,0) {\small $1$};
      %%
      \node[anchor=east] at (4,\Hg) {\small \textbf{Index} $=1$};
      \node[anchor=east]  at (4,\Hf) {\small $2$};
      \node[anchor=east]  at (4,\He) {\small $3$};
      \node[anchor=east]  at (4,\Hd) {\small $4$};
      \node[anchor=east]  at (4,\Hc) {\small $6$};
      \node[anchor=east]  at (4,\Hb) {\small $8$};
      \node[anchor=east]  at (4,\Ha) {\small $12$};
      \node[anchor=east]  at (4,0) {\small $24$};
    \end{scope}
  \end{tikzpicture}
  \]

    \[
  \begin{tikzpicture}[scale=.85, shorten >= -3pt, shorten <= -3pt]
     \tikzstyle{f} = [faded]
    \begin{scope}[scale=1,shift={(0,0)},xscale=1.1]
      \tikzstyle{every node}=[font=\footnotesize]
      \node (G) at (-1.75,8.8) {$Q_8\rtimes C_9$};
      \node (Q8xC3) at (-3.5,6.5) {$Q_8\times C_3$};
      \node (C18-1) at (2.8,5.8) {$C_{18}$}; \node (C18-2) at (2.1,5.8) {$C_{18}$};
      \node (C18-3) at (1.4,5.8) {$C_{18}$}; \node (C18-4) at (.7,5.8) {$C_{18}$};
      \node (C12-1) at (-1.05,4.5+.5) {$C_{12}$}; \node (C12-2) at (-1.75,4.5+.5) {$C_{12}$};
      \node (C12-3) at (-2.45,4.5+.5) {$C_{12}$};
      \node (C9-1) at (4.55,4.3) {$C_9$}; \node (C9-2) at (3.85,4.3) {$C_9$};
      \node (C9-3) at (3.15,4.3) {$C_9$}; \node (C9-4) at (2.45,4.3) {$C_9$};
      \node (Q8) at (-5.25,4.5-.5) {$Q_8$};
      \node (C6) at (0,3+.5) {$C_6$};
      \node (C4-1) at (-2.8,3-.5) {$C_4$}; \node (C4-2) at (-3.5,3-.5) {$C_4$}; \node (C4-3) at (-4.2,3-.5) {$C_4$};
      \node (C3) at (1.75,1.5+.5) {$C_3$};
      \node (C2) at (-1.75,1.5-.5) {$C_2$};
      \node (1) at (0,0-.5) {$C_1$};
      \draw[f] (Q8xC3) to (C12-1); \draw[f] (Q8xC3) to (C12-2); \draw[f] (Q8xC3) to (C12-3); 
      \draw[f] (C12-1) to (C6); \draw[f] (C12-2) to (C6); \draw[f] (C12-3) to (C6);
      \draw[f] (C6) to (C3); \draw[f] (C3) to (1);
      \draw[f] (Q8xC3) to (Q8); 
      \draw[f] (Q8) to (C4-1); \draw[f] (Q8) to (C4-2); \draw[f] (Q8) to (C4-3);
      \draw[f] (C4-1) to (C2); \draw[f] (C4-2) to (C2); \draw[f] (C4-3) to (C2);
      \draw[f] (C2) to (1);
      \draw[f] (C12-1) to (C4-1); \draw[f] (C12-2) to (C4-2);  \draw[f] (C12-3) to (C4-3); 
      \draw[f] (C6) to (C2);
      \draw[f] (C9-1) to (C3); \draw[f] (C9-2) to (C3); \draw[f] (C9-3) to (C3); \draw[f] (C9-4) to (C3); 
      \draw[f] (C18-1) to (C6); \draw[f] (C18-2) to (C6); \draw[f] (C18-3) to (C6); \draw[f] (C18-4) to (C6); 
      \draw[f] (C18-1) to (C9-1); \draw[f] (C18-2) to (C9-2); 
      \draw[f] (C18-3) to (C9-3); \draw[f] (C18-4) to (C9-4);
      \draw[f] (G) to (C18-1); \draw[f] (G) to (C18-2); \draw[f] (G) to (C18-3); \draw[f] (G) to (C18-4); 
      \draw[f] (G) to (Q8xC3);
      %%
      \node[anchor=east] at (-7,8.8) {\small \textbf{Order} $=72$};
      \node[anchor=east] at (-7,6.5) {\small $24$};
      \node[anchor=east]  at (-7,5.8) {\small $18$};
      \node[anchor=east]  at (-7,5) {\small $12$};
      \node[anchor=east]  at (-7,4.35) {\small $9$};
      \node[anchor=east]  at (-7,3.95) {\small $8$};
      \node[anchor=east]  at (-7,3.5) {\small $6$};
      \node[anchor=east]  at (-7,2.5) {\small $4$};
      \node[anchor=east]  at (-7,2) {\small $3$};
      \node[anchor=east]  at (-7,1) {\small $2$};
      \node[anchor=east]  at (-7,0) {\small $1$};
      %%
      \node[anchor=east] at (6,8.8) {\small \textbf{Index} $=1$};
      \node[anchor=east] at (6,6.5) {\small $3$};
      \node[anchor=east]  at (6,5.8) {\small $4$};
      \node[anchor=east]  at (6,5) {\small $6$};
      \node[anchor=east]  at (6,4.35) {\small $8$};
      \node[anchor=east]  at (6,3.95) {\small $9$};
      \node[anchor=east]  at (6,3.5) {\small $12$};
      \node[anchor=east]  at (6,2.5) {\small $18$};
      \node[anchor=east]  at (6,2) {\small $24$};
      \node[anchor=east]  at (6,1) {\small $36$};
      \node[anchor=east]  at (6,0) {\small $72$};
    \end{scope}
   \end{tikzpicture}
   \]

\item Given a group $G$, prove the following elementary properties about its
  commutator subgroup, $G'=\<xyx^{-1}y^{-1}\mid x,y\in
  G\>$. 
  \begin{enumerate}
  \item $G'\normaleq G$ and $G/G'$ is abelian.
  \item If $G'\leq H\leq G$, then $H\normaleq G$.
  \item If $N\normaleq G$, then $N'\normaleq G$. 
  \item If $N\normaleq G$, then $G/N$ is abelian if and only if
    $G'\normaleq N$.
  \item If $\phi\colon G_1\to G_2$ is a homomorphism, then
    \[
    \phi([h,k])=[\phi(h),\phi(k)]\qquad\text{and}\qquad
    \phi([H,K])=[\phi(H),\phi(K)],
    \]
    for all $h,k\in G_1$, and $H,K\leq G_1$.
  \end{enumerate}

  
\end{enumerate}

\end{document}
