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\noindent
\parbox{2.25in}{\bf Math 3110-101}
\hfill {\Large \bf Homework \#5} \hfill
\parbox{2.25in}{\bf \hfill Due: Wed., Oct. $7^\mathrm{th}$, 2020}

\vspace{0.2in}

\begin{enumerate}
\item 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 $(12)(34)(56)(78)$.

\item The following pairs of groups are isomorphic. Prove it.
\begin{enumerate}[(a)]
\item $U(7)$ and $\mathbb{Z}_6$
\item $\displaystyle H = \left\{ \begin{bmatrix} 1 & n \\ 0 & 1 \end{bmatrix} \; \Bigg| \; n \in \mathbb{Q} \right\}$ and $\mathbb{Q}$
\end{enumerate}

\item The following pairs of groups are \underline{\bf not} isomorphic. Prove it.
\begin{enumerate}[(a)]
\item $(\mathbb{Z}_5)^{2 \times 2}$ and $\mathrm{GL}_2(\mathbb{R})$
\item $\mathbb{Z}_{222}$ and $D_{111}$
\item $A_4$ and $D_6$
\item $\mathbb{R}_{\not=0}$ and $\mathbb{C}_{\not=0}$ (non-zero reals and complex numbers both under multiplication)
\end{enumerate}

\item 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 $G_1$.
\end{enumerate}

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