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\parbox{1.95in}{\bf Math 3110}
\hfill {\Large \bf Homework \#6} \hfill
\parbox{1.95in}{\bf \hfill Due: \underline{\bf Wed., Mar. 5${}^{\text{\bf th}}$}\!, 2025}

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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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\begin{description}
\item[\#1] Cayley's theorem tells us that $Q = \{ \pm 1, \pm i, \pm j, \pm k \}$ is isomorphic to a subgroup of $S_8$. Find such a subgroup using the ordering of elements: $1,-1,i,-i,j,-j,k,-k$ (so, for example, $i$ is element \#3). To help you get started, left multiplication by $-1$ sends $1$ to $-1$, $-1$ to $1$, $i$ to $-i$, etc. so it sends 1 to 2, 2 to 1, 3 to 4, etc. Thus $-1$ corresponds with $(12)(34)(56)(78)$.

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\item[\#2] The following pairs of groups are isomorphic. Prove it.
\begin{enumerate}[(a)]
\item $U(7)$ and $\mathbb{Z}_6$

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

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\item $\displaystyle H = \left\{ \begin{bmatrix} 1 & n \\ 0 & 1 \end{bmatrix} \; \Bigg| \; n \in \mathbb{Q} \right\}$ and $\mathbb{Q}$ 

{\it Note:} You need to come up with an isomorphism, $\varphi:H \to \mathbb{Q}$, and prove it is an isomorphism.
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\item[\#3] 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 $\mathbb{Z}_{246}$ and $D_{123}$

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\item $(\mathbb{Z}_{99})^{2 \times 2}$ and $\mathrm{GL}_2(\mathbb{Z})$

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\item $A_4$ and $D_6$

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\item $\mathbb{R}_{\not=0}$ and $\mathbb{C}_{\not=0}$ (non-zero reals and complex numbers both under multiplication)
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\item[\#4] Let $\varphi:G_1 \to G_2$ be an isomorphism
\begin{enumerate}[(a)]
\item Show $\varphi(\langle g \rangle) = \langle \varphi(g) \rangle$ for any $g \in G_1$. [This implies that $G_1$ is cyclic iff $G_2$ is cyclic.]
\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 $G_1$.
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