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\noindent
\parbox{1.95in}{\bf Math 3110}
\hfill {\Large \bf Homework \#6} \hfill
\parbox{1.95in}{\bf \hfill Due: For Practice} %\underline{\bf Wed., Oct. 11${}^{\text{\bf th}}$}, 2023}

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%\noindent {\bf\large Name:} \quad \underline{\large\qquad This is my Name \qquad}

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%Please remember when submitting any work via email or in person to\dots\\
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\noindent
{\bf Note:} Due to the Hurricane. We will forego turning this assignment in for a grade. You need to know this material. It would be good to work through these problems yourselves, but I will just provide an answer key via email. \underline{\bf You do not need to turn this problem set in.}

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\begin{description}
\item[\#1] The following pairs of groups are \underline{\bf not} isomorphic. Prove it.

{\it Hint/Note:} Find a (group) property that does not match.
\begin{enumerate}[(a)]
\item $\mathbb{Z}^{3 \times 3}$ and $\mathrm{GL}_3(\mathbb{Z})$

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\item $U(30)$ and $Q=\{\pm 1, \pm i, \pm j, \pm k\}$ (the quaternion group)

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\item $S_5$ and $D_{60}$

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\item $\mathbb{Q}$ and $\mathbb{Z}$ 

This one is a little tricky. {\it Hint:} If $0 \not= \frac{a}{b} \in \mathbb{Q}$, then $\frac{a}{2b} \not\in \langle \frac{a}{b} \rangle$.
\end{enumerate}

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\item[\#2] The following pairs of groups are isomorphic. Prove it.
\begin{enumerate}[(a)]
\item $U(10)$ and $\mathbb{Z}_4$

{\it Hint:} You could cite a theorem and prove this in one line.

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\item $\displaystyle G = \left\{ \begin{bmatrix} a & -b \\ b & a \end{bmatrix} \; \Bigg| \; a,b \in \mathbb{R} \right\}$ and $\mathbb{C}$ 

{\it Notes:} \\
\force \quad $\bullet$ You may assume we already know $G$ is a group (of $2\times 2$ real matrices) under addition.\\
\force \quad $\bullet$ You need to come up with an isomorphism, $\varphi:G \to \mathbb{C}$, and prove it is an isomorphism.

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\item[\#3] Let $\varphi:G_1 \to G_2$ be an isomorphism
\begin{enumerate}[(a)]
\item Let $b \in G_1$ and $k \in \mathbb{Z}$. Explain why $x^k=b$ (in $G_1$) and $x^k=\varphi(b)$ (in $G_2$) must have the same number of solutions.
\item Let $K$ be a subgroup of $G_2$. Show that $\varphi^{-1}(K) = \{ g \in G_1 \;|\; \varphi(g) \in K\}$ is a subgroup of $G_1$.
\end{enumerate}

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\item[\#4] Cayley's theorem tells us that $D_4 = \langle x,y \;|\; x^4=1, y^2=1, (xy)^2=1 \rangle = \{1,x,x^2,x^3,y,xy,x^2y,x^3y\}$ is isomorphic to a subgroup of $S_8$. Find such a subgroup using the ordering of elements: $1,x,x^2,x^3,y,xy,x^2y,x^3y$ (so, for example, $x^3$ is element \#4 and $x^2y$ is element \#7). To help you get started, left multiplication by $x$ sends $1$ to $x$, $x$ to $x^2$, $x^2$ to $x^3$, $x^3$ to $1$, $y$ to $xy$, etc. so it sends 1 to 2, 2 to 3, 3 to 4, 4 to 1, 5 to 6 etc. Thus $x$ corresponds with $(1234)(5678)$.

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{\it Note:} Your answer should be a set of permutations: $\{(1),(1234)(5678), \dots \}$


 
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