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\pagestyle{fancy} \lhead{{\sf Homework 2 $|$ Due January 31 (Monday)
}} \rhead{\thepage} \cfoot{{\sf Math 4120/6120 $|$ Modern Algebra $|$
    Spring 2022 $|$ M.~Macauley}}

\begin{document}

%$\;$

\begin{enumerate}

  
\item For each $n$, sketch the $n^{\rm th}$ roots of unity on the unit
  circle, and list the primitive $d^{\rm th}$ roots for each
  $d\mid n$. Then factor $x^n-1$ as a product of irreducible
  polynomials. \vspace{-2mm}
  \begin{enumerate}
    \begin{multicols}{4}
    \item $n=8$
    \item $n=9$
    \item $n=10$
    \item $n=16$.
    \end{multicols}
  \end{enumerate}

  \item For each $n$ from the previous problem, the set $U_n:=\{k\mid 0<
    k<n,\;\gcd(n,k)=1\}$ forms a group under multiplication, where the
    result is taken modulo $n$. Construct a Cayley table, Cayley
    diagram, and determine to which familiar group it is isomorphic.


%%---------------------------------------------------------------------
  
\item Below are Cayley diagrams of the \emph{generalized quaternion
  group} $Q_{16}\!=\!\<\zeta_8,j\>$, defined by replacing
  $\zeta_4=e^{2\pi i/4}=i$ with $\zeta_8=e^{2\pi
  i/8}=\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i$ in the quaternion group $Q_8$.

  \vspace{-3mm}

\tikzstyle{v-small} = [circle, draw, fill=lightgrey,inner sep=0pt, 
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\tikzstyle{v-tiny} = [circle, draw, fill=lightgrey,inner sep=0pt, 
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\tikzstyle{R} = [draw, very thick, Red,-stealth,bend right=10]
\tikzstyle{R-out} = [draw, very thick, Red,-stealth,bend right=15]
\tikzstyle{R-in} = [draw, very thick, Red,-stealth,bend left=12]
\tikzstyle{B} = [draw, very thick, Blue,-stealth,bend right=35]
\tikzstyle{B2} = [draw, very thick, Blue,-stealth,bend right=45]
\[
  \begin{tikzpicture}[scale=1.27,auto]
   \tikzstyle{every node}=[font=\scriptsize]
    \begin{scope}[shift={(0,0)},scale=1,auto]
    \tikzstyle{every node}=[font=\scriptsize]
      \node (s) at (0:1) [v-small] {};
      \node (rs) at (45:1) [v-small] {};
      \node (r2s) at (90:1) [v-small] {};
      \node (r3s) at (135:1) [v-small] {};
      \node (r4s) at (180:1) [v-small] {};
      \node (r5s) at (225:1) [v-small] {};
      \node (r6s) at (270:1) [v-small] {};
      \node (r7s) at (315:1) [v-small] {};
      %%
      \node (1) at (0:2) [v-small] {$1$};
      \node (r) at (45:2) [v-small] {};
      \node (r2) at (90:2) [v-small] {};
      \node (r3) at (135:2) [v-small] {};
      \node (r4) at (180:2) [v-small] {};
      \node (r5) at (225:2) [v-small] {};
      \node (r6) at (270:2) [v-small] {};
      \node (r7) at (315:2) [v-small] {};
      %%
      \draw [b] (1) to (s); \draw [B] (s) to (r4);
      \draw [b] (r4) to (r4s); \draw [B] (r4s) to (1);
      \draw [b] (r) to (rs); \draw [B] (rs) to (r5);
      \draw [b] (r5) to (r5s); \draw [B] (r5s) to (r);
      \draw [b] (r2) to (r2s); \draw [B] (r2s) to (r6);
      \draw [b] (r6) to (r6s); \draw [B] (r6s) to (r2);
      \draw [b] (r3) to (r3s); \draw [B] (r3s) to (r7);
      \draw [b] (r7) to (r7s); \draw [B] (r7s) to (r3);
      %%
      \draw [R-out] (1) 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 (r5);
      \draw [R-out] (r5) to (r6);
      \draw [R-out] (r6) to (r7);
      \draw [R-out] (r7) to (1);
      %%
      \draw [R-in] (s) to (r7s);
      \draw [R-in] (r7s) to (r6s);
      \draw [R-in] (r6s) to (r5s);
      \draw [R-in] (r5s) to (r4s);
      \draw [R-in] (r4s) to (r3s);
      \draw [R-in] (r3s) to (r2s);
      \draw [R-in] (r2s) to (rs);
      \draw [R-in] (rs) to (s);
      %%
    \end{scope}   
  \begin{scope}[shift={(6.75,0)},scale=.5]
    \node[v-small] (1) at (90:4) {};
    \node[v-small] (r) at (90:3) {};
    \node[v-small] (r2) at (90:2) {};
    \node[v-small] (r3) at (90:1) {};
    %%
    \node[v-small] (s) at (0:4) {\scriptsize $1$};
    \node[v-small] (rs) at (0:3) {};
    \node[v-small] (r2s) at (0:2) {};
    \node[v-small] (r3s) at (0:1) {};
    %%
    \node[v-small] (s2) at (-90:4) {};
    \node[v-small] (rs2) at (-90:3) {};
    \node[v-small] (r2s2) at (-90:2) {};
    \node[v-small] (r3s2) at (-90:1) {};
    %%
    \node[v-small] (s3) at (180:4) {};
    \node[v-small] (rs3) at (180:3) {};
    \node[v-small] (r2s3) at (180:2) {};
    \node[v-small] (r3s3) at (180:1) {};
    %%
    \draw[r] (r) to (1);
    \draw[r] (r2) to (r);
    \draw[r] (r3) to (r2);
    %%
    \draw[r] (rs2) to (s2);
    \draw[r] (r2s2) to (rs2);
    \draw[r] (r3s2) to (r2s2);
    %%
    \draw[r] (r2s) to (r3s);
    \draw[r] (rs) to (r2s);
    \draw[r] (s) to (rs);
    %%
    \draw[r] (r2s3) to (r3s3);
    \draw[r] (rs3) to (r2s3);
    \draw[r] (s3) to (rs3);
    %%
    \draw[r,bend right=25] (s2) to (r3);
    \draw[r,bend right=25] (1) to (r3s2); 
    \draw[r,bend right=25] (r3s3) to (s); 
    \draw[r,bend right=25] (r3s) to (s3); 
    %%
    \draw[b,bend right=20] (s) to (1);
    \draw[b,bend right=20] (rs) to (r);
    \draw[b,bend right=20] (r2s) to (r2);
    \draw[b,bend right=20] (r3s) to (r3);
    %%
    \draw[b,bend right=20] (s2) to (s);
    \draw[b,bend right=20] (rs2) to (rs);
    \draw[b,bend right=20] (r2s2) to (r2s);
    \draw[b,bend right=20] (r3s2) to (r3s);
    %%
    \draw[b,bend right=20] (s3) to (s2);
    \draw[b,bend right=20] (rs3) to (rs2);
    \draw[b,bend right=20] (r2s3) to (r2s2);
    \draw[b,bend right=20] (r3s3) to (r3s2);
    %%
    \draw[b,bend right=20] (1) to (s3);
    \draw[b,bend right=20] (r) to (rs3);
    \draw[b,bend right=20] (r2) to (r2s3);
    \draw[b,bend right=20] (r3) to (r3s3);
\end{scope}      
  \end{tikzpicture}
  \]
  
  \vspace{-5mm}

  \begin{enumerate}
  \item Draw these diagrams and label each node as $a+bi+cj+dk$. Then
    re-draw them with each node labeled as either $\pm\zeta^m$ or
    $\pm\zeta^mj$, where $\zeta=\zeta_8$ and $m=0,1,2,3$.
  \item Identifying elements of $Q_{16}$ with their negatives defines
    a group on $8$ elements:
    \[
    \pm 1,\;\;\pm\zeta,\;\;\pm\zeta^2,\;\;\pm\zeta^3,\;\;
    \pm j,\;\;\pm\zeta j,\;\;\pm\zeta^2j,\;\;\pm\zeta^3j.
    \]
    \vspace{-1mm} Construct a Cayley table and Cayley diagram. Which
    familiar group is this?
  \end{enumerate}

  %%---------------------------------------------------------------------
  
\item For each part below, the two matrices given generate a group
  $G=\<A,B\>$, where the binary operation is matrix multiplication.
  Draw a Cayley diagram for each group, write a presentation, and
  determine to which familiar group is it isomorphic.
 \begin{enumerate}
    \begin{multicols}{2}
      \item $A=\begin{bmatrix}0&1&0\\1&0&0\\0&0&1\end{bmatrix}$,
        $B=\begin{bmatrix}1&0&0\\0&0&1\\0&1&0\end{bmatrix}$.
      \item $A=\begin{bmatrix}0&i\\i&0\end{bmatrix}$,
        $B=\begin{bmatrix}0&-1\\-1&0\end{bmatrix}$.
      \item $A=\begin{bmatrix}i&0\\0&-i\end{bmatrix}$,
        $B=\begin{bmatrix}0&-1\\1&0\end{bmatrix}$.
      \item $A=\begin{bmatrix}e^{2\pi i/8}&0\\0&e^{-2\pi i/8}\end{bmatrix}$, 
        $B=\begin{bmatrix}0&-1\\1&0\end{bmatrix}$.
    \end{multicols}
 \end{enumerate}

 %%---------------------------------------------------------------------
 
\item For the numbers below, list all abelian groups of that order by
  writing each one as a product of cyclic groups of prime power
  order. Then, determine which group it is isomorphic to of the form
  $\Z_{n_1}\times\cdots\times\Z_{n_k}$, where $n_{i+1}\mid n_i$.\vspace{-2mm}
  
  \begin{enumerate}
    \begin{multicols}{4}
    \item $32=2^5$
    \item $36=2^2\cdot 3^2$
    \item $400=2^4\cdot 5^2$
    \item $p^3q$; primes $p\neq q$ 
    \end{multicols}
  \end{enumerate}

  %%----------------------------------------------------------
  
\end{enumerate}

\end{document}

