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\begin{document}

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\noindent
\parbox{2.25in}{\bf Math 3110-101}
\hfill {\Large \bf Homework \#1} \hfill
\parbox{2.25in}{\bf \hfill Due: Mon., Aug. $31^\mathrm{st}$, 2015}

\vspace{0.2in}

\begin{enumerate}
\item Determine which of the following sets with operations are groups. If it is a group, state its identity, what the inverse of a typical element looks like, and determine if it is Abelian. If it is not a group, state which axioms hold and give counter-examples for those which fail (don't forget closure).
   \begin{enumerate}
   \item $(\mathbb{Z}_{\geq 0},+)$ non-negative integers with addition
   \item $(3\mathbb{Z},+)$ multiples of 3 (i.e. $0,\pm 3,\pm 6, \dots$) with addition
   \item $(\mathbb{R}_{<0},\;\cdot\;)$ negative reals with multiplication
   \item $(\mathbb{R}_{\not= 0}, \div )$ non-zero reals with division
   \item $(\mathbb{Q}_{>0},\;\cdot\;)$ positive rationals with multiplication
   \end{enumerate}
   
\vspace{0.25in}

\item Let $G$ be a group with identity $e \in G$. Suppose that $g^2=e$ for all $g \in G$. 
   \begin{enumerate}
   \item What can be said about inverses of elements in $G$? Orders of elements?
   \item Prove that $G$ must be abelian.
   \end{enumerate}
   
\vspace{0.25in}

\item Consider the dihedral group $D_5 = \{ R_{0^\circ},R_{72^\circ},R_{144^\circ},R_{216^\circ},R_{288^\circ},V_1,V_2,V_3,V_4,V_5 \}$ (symmetries of a regular pentagon). [Rotations are done counter-clockwise and reflections are labeled in the picture below.]

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\draw (-0*360/5:1) node[vertex] (v0) {$\,$};
\draw (-1*360/5:1) node[vertex] (v1) {$\,$};
\draw (-2*360/5:1) node[vertex] (v2) {$\,$};
\draw (-3*360/5:1) node[vertex] (v3) {$\,$};
\draw (-4*360/5:1) node[vertex] (v4) {$\,$};

\draw (-0*360/10:2) node[vertex] (w0) {$\,$};
\draw (-1*360/10:1.5) node[vertex] (w1) {$\,$};
\draw (-2*360/10:2) node[vertex] (w2) {$\,$};
\draw (-3*360/10:1.5) node[vertex] (w3) {$\,$};
\draw (-4*360/10:2) node[vertex] (w4) {$\,$};
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\draw (-8*360/10:2) node[vertex] (w8) {$\,$};
\draw (-9*360/10:1.5) node[vertex] (w9) {$\,$};

\draw (-0*360/5:2.5) node (l0) {$V_1$};
\draw (-1*360/5:2.5) node (l1) {$V_2$};
\draw (-2*360/5:2.5) node (l2) {$V_3$};
\draw (-3*360/5:2.5) node (l3) {$V_4$};
\draw (-4*360/5:2.5) node (l4) {$V_5$};

\draw (v0) -- (v1);
\draw (v1) -- (v2);
\draw (v2) -- (v3);
\draw (v3) -- (v4);
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\draw [dotted] (w0)--(w5);
\draw [dotted] (w1)--(w6);
\draw [dotted] (w2)--(w7);
\draw [dotted] (w3)--(w8);
\draw [dotted] (w4)--(w9);

\draw[->,thick] (1.5,-1.75) arc (-55:0:1.25) ;
\draw (1.75,-2.75) node (rot) {$R_{0^\circ},R_{72^\circ},R_{144^\circ},\dots$};

\end{tikzpicture}
\hfill \force

\begin{enumerate}
\item Compute $V_1R_{72^\circ}$, $R_{144^\circ}V_3$, and $V_2V_5$.
\item Is $D_5$ Abelian? Why or why not?
\item Find the inverse of each element ($R_{0^\circ}^{-1} = ???$, $R_{72^\circ}^{-1} = ???$, etc.).
\item Find the order of each element.
\end{enumerate}

\end{enumerate}

\end{document}