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\begin{document}

\begin{center}
	\Large{Math 31 - Homework 3}
	
	\vspace{2mm}
	\large{Due Wednesday, July 10}
\end{center}

\paragraph{Note:} Any problem labeled as ``show'' or ``prove'' should be written up as a formal proof, using complete sentences to convey your ideas.

\subsection*{Easier}

\paragraph{1.} Let $D_4$ be the 4th dihedral group, which consists of symmetries of the square. Let $r \in D_4$ denote counterclockwise 
rotation by $90^\circ$, and let $m$ denote reflection across the vertical axis.
\begin{center}
\begin{tikzpicture}
	\draw [black,line width=1pt] (0,0) node [left]{$1$} -- (2,0) node [right]{$2$} -- (2,2) node [right]{$3$} -- (0,2) node [left]{$4$} -- cycle;
	\draw [->,thick] (3,1) -- (5, 1);
	\draw [black,line width=1pt] (6,0) node [left]{$4$} -- (8,0) node [right]{$1$} -- (8,2) node [right]{$2$} -- (6,2) node [left]{$3$} -- cycle; 
	\node [above] at (4,1.1) {$r$};
\end{tikzpicture}
\end{center}
\vspace{2mm}
\begin{center}
\begin{tikzpicture}
	\draw [black,line width=1pt] (0,0) node [left]{$1$} -- (2,0) node [right]{$2$} -- (2,2) node [right]{$3$} -- (0,2) node [left]{$4$} -- cycle;
	\draw [->,thick] (3,1) -- (5, 1);
	\draw [black,line width=1pt] (6,0) node [left]{$2$} -- (8,0) node [right]{$1$} -- (8,2) node [right]{$4$} -- (6,2) node [left]{$3$} -- cycle; 
	\node [above] at (4,1.1) {$m$};
\end{tikzpicture}
\end{center}
Check that
\[
	rm = mr^{-1}.
\]
Conclude that $D_4$ is a nonabelian group of order 8.

\paragraph{2.} We mentioned in class that elements of $D_n$ can be thought of as permutations of the vertices of the regular $n$-gon. For example, the rotation 
$r$ of the square mentioned in the last problem can be identified with the permutation
\[
	\rho = \left( \begin{array}{cccc}
		1 & 2 & 3 & 4 \\
		2 & 3 & 4 & 1
	\end{array} \right).
\]
Write the reflection $m$ as a permutation $\mu \in S_4$, and compute the product $\rho \mu$ in $S_4$. Then compute $rm \in D_4$, and write it as a permutation 
$\sigma$. Check that $\sigma = \rho \mu$. (In other words, this identification of symmetries of the square with permutations respects the group operations.)

\paragraph{3.} Recall that if $*$ is a binary operation on a set $S$, an element $x$ of $S$ is an \textbf{idempotent} if $x*x=x$. Prove that a group has exactly one 
	idempotent element.
	
\paragraph{4.} Consider the group $\la \Z_{30}, +_{30} \ra$ under addition.
\begin{thmenum}
	\item Find the orders of the elements 3, 4, 6, 7, and 18 in $\Z_{30}$.
	\item Find all the generators of $\la \Z_{30}, +_{30} \ra$.
\end{thmenum}

\paragraph{5.} Determine whether each of the following subsets is a subgroup of the given group. If not, state which of the subgroup axioms fails.
\begin{thmenum}
	\item The set of real numbers $\R$, viewed as a subset of the complex numbers $\C$ (under addition).
	\item The set $\pi \Q$ of rational multiples of $\pi$, as a subset of $\R$.
	\item The set of $n \times n$ matrices with determinant 2, as a subset of $\gln(\R)$.
	\item The set $\{i, m_1, m_2, m_3\} \subset D_3$ of reflections of the equilateral triangle, along with the identity transformation.
\end{thmenum}

\subsection*{Medium}
\paragraph{6.} [Saracino, Section 4, \#25] Show that if $G$ is a finite group and $\abs{G}$ is even, then there is an element $a \in G$ such that $a \neq e$ and 
$a^2=e$.

\paragraph{7.} [Saracino, Section 4, \#21] Let $a$ and $b$ be elements of a group $G$. Show that if $ab$ has finite order $n$, then $ba$ also has order $n$.

\paragraph{8.} [Saracino, Section 4, \#20] Let $G$ be a group and let $a \in G$. An element $b \in G$ is called a \emph{conjugate} of $a$ if there exists an element 
$x \in G$ such that $b = xax^{-1}$. Show that any conjugate of $a$ has the same order as $a$.

\paragraph{9.} Let $G$ be a group. If $H$ and $K$ are subgroups of $G$, show that $H \cap K$ is also a subgroup of $G$.

\paragraph{10.} Let $r$ and $s$ be positive integers, and define 
\[
	H = \left\{nr + ms : n, m \in \Z \right\}.
\]
\begin{thmenum}
	\item Show that $H$ is a subgroup of $\Z$.
	\item We saw in class that every subgroup of $\Z$ is cyclic. Therefore, $H = \la d \ra$ for some $d \in \Z$. What is this integer $d$? Prove that the 		$d$ you've found is in fact a generator for $H$.
\end{thmenum}












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