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\begin{center}
{\large MATH 150A Winter 2020 \ -  \ {\bf Problem Set 8}}

% Put your name and section here.
{\large due March 6}
\end{center}


\begin{enumerate}

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	\item Draw the Cayley graph for each group and generating set.
	\begin{enumerate}
	 \item $C_{10}$ generated by $\{x\}$.
	 \item $C_{10}$ generated by $\{x^2,x^5\}$.
	 \item $A_4$ generated by $\{(1\;2\;3),(2\;3\;4)\}$.
	 \item $C_2 \times C_2 \times C_2$ generated by $\{(x,1,1),(1,x,1),(1,1,x)\}$.
	\end{enumerate}

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	\item Let $G$ be a group generated by $S$ and $H$ a subgroup of $G$ generated by $T \subseteq S$.  Prove that $H$ is normal in $G$ if and only if all edges labelled by elements of $T$ are loops in the Schreier coset graph of $H$ in $G$ with generating set $S$.

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	\item Given two elements of the lamplighter group
	\[ g = (n,(\ldots,l_{-1},l_0,l_1,\ldots)), \]
	\[ h = (m,(\ldots,k_{-1},k_0,k_1,\ldots)), \]
	how can one determine if they are conjugates?
	
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	\item The {\em infinite dihedral group} $D_\infty$ is a subgroup of permutations of the integers generated by $f(n) = -n$ and $g(n) = 1-n$, which reflect the integer number line over the point 0 and 1/2 respectively.
	 \begin{enumerate}
	 \item Give a presentation of $D_\infty$.
	 \item Demonstrate a surjective homomorphism to each finite dihedral group $\varphi:D_\infty\to D_n$ for $n \geq 3$.
	 \end{enumerate}

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	\item Use the Todd-Coxeter algorithm to analyze the group generated $\{x,y\}$ with the following relations.  Determine the order of the group and identify the group if you can.
	\begin{enumerate}
	 \item $x^2 = 1$, $y^2 = 1$, $xyx = yxy$,
	 \item $x^3 = 1$, $y^3 = 1$, $xyx = yxy$,
	 \item $x^4 = 1$, $y^2 = 1$, $xyx = yxy$,
	 \item $x^4 = 1$, $y^4 = 1$, $x^2y^2 = 1$,
	 \item $x^3 = 1$, $y^2 = 1$, $yxyxy = 1$,
	 \item $x^3 = 1$, $y^3 = 1$, $yxyxy = 1$.
	\end{enumerate}

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\end{enumerate}
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