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\pagestyle{fancy} \lhead{{\sf Homework 1 $|$ Due September 3 (Monday)
}} \rhead{\thepage} \cfoot{{\sf Math 8510 $|$ Abstract Algebra I  $|$
    Fall 2018 $|$ M.~Macauley}}

\begin{document}

$\;$ \vspace{-3mm}


\begin{enumerate}

\item Consider the frieze pattern below at left, with the diamonds labeled for reference.
  \tikzstyle{up}=[shape=diamond, orange,
    aspect=.5,anchor=south,fill=orange,draw]
  \tikzstyle{down}=[shape=diamond, orange,
    aspect=.5,anchor=north,fill=orange,draw]
  \colorlet{lightgrey}{black!15}
  \tikzstyle{vert} = [circle=2pt, draw, fill=lightgrey]
  \tikzstyle{B} = [draw,very thick, blue]
  \tikzstyle{R} = [draw, very thick, Red,-stealth']

  \begin{tikzpicture}[scale=1.05,baseline=(current bounding box.center)]
    \begin{scope}[scale=.65,shift={(0,0)}] % glide and horizontal flip
    \draw (0,0) node [up] {};
    \draw (1,0) node [down] {};
    \draw (2,0) node [up] {};
    \draw (3,0) node [down] {};
    \draw (4,0) node [up] {};
    \draw (5,0) node [down] {};
    \draw (6,0) node [up] {};
    \draw (7,0) node [down] {};
    \draw (8,0) node [up] {};
    \node at (0,1.5) {$-4$};
    \node at (1,-1.5) {$-3$};
    \node at (2,1.5) {$-2$};
    \node at (3,-1.5) {$-1$};
    \node at (4,1.5) {$0$};
    \node at (5,-1.5) {$1$};
    \node at (6,1.5) {$2$};
    \node at (7,-1.5) {$3$};
    \node at (8,1.5) {$4$};
    \draw (-1,0) -- (9,0);
    \draw (-1.5,0) node {$\cdots$};
    \draw (9.5,0) node {$\cdots$};
    \draw[dotted] (2,-1.75)--(2,1.75);
    \draw[dotted] (3,-1.75)--(3,1.75);
    \draw[dashed,Red,thick] (4,-2)--(4,2);
    \draw[dotted] (5,-1.75)--(5,1.75);
    \draw[dotted] (6,-1.75)--(6,1.75);
    \end{scope}
    \begin{scope}[scale=.9,shift={(9.5,-.7)}]
      \foreach \x in {0,1.5,...,5} {
        \node[vert] at (\x,0) {};
        \node[vert] at (\x,1.5) {};
        \path[B] (\x, .15) -- (\x, 1.35);
      }
      \foreach \x in {0,1.5,...,4} {
        \path[R] (\x+1.35, 0) -- (\x+.15, 0);
        \path[R] (\x+.15, 1.5) -- (\x+1.35, 1.5);
      }
      \path[R] (-.15, 0) -- (-.85, 0);
      \path[R] (-.85, 1.5) -- (-.15, 1.5);
      \path[R] (5.4, 0) -- (4.7, 0);
      \path[R] (4.7, 1.5) -- (5.4, 1.5);
      \draw (-1.1,0) node[red] {$\cdots$};  
      \draw (-1.1,1.5) node[red] {$\cdots$};
      \draw (5.75,0) node[red] {$\cdots$};
      \draw (5.75,1.5) node[red] {$\cdots$};
    \end{scope}
  \end{tikzpicture}

  Let $g$ be a glide-reflection to the right, and $h$ a horizontal
  flip about the dashed red line. These actions generate the symmetry group of this frieze, and its Cayley diagram is shown on the right. For each of the four dotted lines shown in the freize, consider the reflection across that line and the resulting re-numbering of the nine diamonds pictured. Then do the following:
\begin{enumerate}
\item Redraw the frieze above and re-number the diamonds accordingly. 
\item Express this reflection in terms of $g$ and $h$.
\end{enumerate}
Write down two presentations for this freize group: one generated by $g$ and $h$, and another by $h$ and the reflection across the diamond numbered $1$. How is this group related to $D_n$?  \\ 



\item Let $G$ be a finite group. 
  \begin{enumerate}
    \item Show that for any $n>2$, the number of elements in $G$ of
      order $n$ is even.
    \item If $|G|$ is even, show that $G$ has an element of order $2$. \\
  \end{enumerate}



\item Let $K\leq H\leq G$ be groups. Show that $[G:K]=[G:H][H:K]$. \\



\item Let $H$ be a subgroup of $G$. 
\begin{enumerate}
\item Show that $gHg^{-1}$ is a subgroup, and that $H\cong gHg^{-1}$.
\item Show that in any group, $|xy|=|yx|$.
\item Show that if $G$ has a unique element $x$ of order 2, then $x\in Z(G)$. \\
\end{enumerate} 



\item Let $|G|=\infty$, and $[G:H]<\infty$. Show that $H$ intersects
  every infinite subgroup of $G$ nontrivially. \\



\item If $A,B\leq G$ and $y\in G$, define the {\em $(A,B)$-double coset}
  $AyB=\{ayb\mid a\in A,\,b\in B\}$.
  \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 $|AyB|=[A^y:A^y\cap B]\cdot |B|$, where $A^y=y^{-1}Ay$.~  \\
  \end{enumerate}


 
\item Prove that if $A$ and $B$ are normal subgroups of $G$ and $AB=G$
  then
  \[
  G/(A\cap B)\cong (G/A)\times(G/B).
  \]


\end{enumerate}

\end{document}

