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\begin{document}

\begin{center}
	\LARGE{Math 31 - Homework 6}
	
	\vspace{2mm}
	\large{Due Friday, August 10}
\end{center}

\subsection*{Easy}
\paragraph{1.} Classify all abelian groups of order 600 up to isomorphism.

\paragraph{2.} Prove that if $G_1$ and $G_2$ are abelian groups, then $G_1 \times G_2$ is abelian.

\paragraph{3.} [Herstein, Section 2.9 \#1] If $G_1$ and $G_2$ are groups, prove that $G_1 \times G_2 \cong G_2 \times G_1$.

\paragraph{4.} [Herstein, Section 4.2 \#2] Let $R$ be an integral domain. If $a,b, c \in R$ with $a \neq 0$ and $ab = ac$, show that $b=c$.

\paragraph{5.} [Herstein, Section 4.1 \#13] Find the following products of quaternions.
	\begin{thmenum}
		\item $(i+j)(i-j)$.
		\item $(1-i+2j-2k)(1+2i-4j+6k)$.
		\item $(2i-3j+4k)^2$.
		\item $i(\alpha_0+\alpha_1i + \alpha_2j + \alpha_3k) - (\alpha_0 + \alpha_1 i + \alpha_2j + \alpha_3k)i$.
	\end{thmenum}
	
\subsection*{Medium}
\paragraph{6.} Let $G_1$ and $G_2$ be finite groups.
	\begin{thmenum}
		\item If $a_1 \in G_1$ and $a_2 \in G_2$, prove that
			\[
				\abs{(a_1, a_2)} = \lcm(\abs{a_1}, \abs{a_2}),
			\]
			where $\lcm(m,n)$ denotes the least common multiple of the integers $m$ and $n$.
		\item {[Herstein, Section 2.9 \#2]} Prove that if $G_1$ is a cyclic group of order $n$ and $G_2$ is a cyclic group of order $m$, then $G_1 \times G_2$ 			is a cyclic group if and only if $\gcd(m,n)=1$. [\textbf{Hint:} $mn = \lcm(mn) \cdot \gcd(m,n)$.] (Note that this implies the theorem that was 				mentioned in class: if $m$ and $n$ are relatively prime, then $\Z_m \times \Z_n \cong \Z_{mn}$.)
	\end{thmenum}
	
\paragraph{7.} [Herstein, Section 4.1 \#4] Recall that if $a \in \Q$, we can always write $a$ in \textbf{lowest terms} (or \textbf{reduced form}) as $a = m/n$, where $\gcd(m,n)=1$. Define
\[
	R = \left\{ \frac{m}{n} \in \Q : \gcd(m,n)=1 \text{ and } n \text{ is odd} \right\}.
\]
That is, $R$ is the set of all rationals which, when written in lowest terms, have an odd denominator. Show that $R$ is a ring under the usual addition and multiplication of rational numbers. Determine which elements of $R$ are units. (This ring $R$ is actually quite important in higher algebra. It is usually denoted by $\Z_{(2)}$, and called the \emph{localization} of $\Z$ at 2.)

\paragraph{8.} [Herstein, Section 4.1 \#21] Show that any field is an integral domain.

\paragraph{9.} [Herstein, Section 4.2 \#3] Let $R$ be a finite integral domain with identity $1 \in R$. Show that $R$ is actually a field.

\subsection*{Hard}
\paragraph{10.} Let $D_n$ denote the dihedral group of order $2n$, let $r \in D_n$ denote the counterclockwise rotation by $2\pi/n$ radians, and let $m$ denote any reflection of the regular $n$-gon. Recall that the rotation subgroup
\[
	H = \left\{e, r, r^2, \ldots, r^{n-1} \right\}
\]
is a normal subgroup of $D_n$. Let $K = \{e, m\}$; we saw in class that $K$ is a subgroup, but it is not normal. You will need two facts regarding $D_n$ (which you do not need to prove):
\begin{enumerate}
	\item Every element of $D_n$ can be written as $r^i m^j$, with $1 \leq i \leq n-1$ and $j=0$ or 1. Thus $D_n = HK$.
	\item You proved earlier (in a special case) that $mr = r^{-1}m$.
\end{enumerate}
Define a group $G$ as follows: the elements of $G$ are pairs $(r^i, m^j)$ (so that $G = H \times K$ \emph{as sets}), but the binary operation on $G$ is ``twisted'' in some sense. More specifically, we define
\begin{align*}
	(r^i, e)(r^j, e) &= (r^{i+j}, e) \\
	(r^i, m)(r^j, e) &= (r^{i-j}, m) \\
	(r^i, e)(r^j, m) &= (r^{i+j}, m) \\
	(r^i, m)(r^j, m) &= (r^{i-j}, e)
\end{align*}
for all $i, j$ between 1 and $n-1$. Now define $\varphi : G \to D_n$ by
\[
	\varphi(r^i, m^j) = r^i m^j.
\]
Prove that $\varphi$ is an isomorphism of $G$ onto $D_n$. (This shows that $D_n$ is not quite a direct product of the subgroups $H$ and $K$, since $K$ isn't normal. However, things can be made to work if we modify the multiplication on $H \times K$ slightly.)















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