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\parbox{1.95in}{\bf Math 3110}
\hfill {\Large \bf Homework \#7} \hfill
\parbox{1.95in}{\bf \hfill Due: {\bf Mon., Mar. 25${}^{\text{\bf th}}$\!, 2024}}

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Please remember when submitting any work via email or in person to\dots\\
\force \hfill \fbox{\fbox{\bf\large\color{red}PUT YOUR NAME ON YOUR WORK!}} \hfill \force

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\begin{description}
\item[\#1] Suppose $H$ and $K$ are subgroups of some group $G$ such that $|H|=66$ and $|K|=42$. Just considering divisibility criteria coming from Lagrange's Theorem, what is the largest size the subgroup $H\cap K$ could be? Suppose $M$ contains both $H$ and $K$. What is the smallest size $M$ could be?

\item[\#2] Direct products.\footnote{Gallian uses $G \oplus H$ for direct products. I will use the more standard $G \times H$ notation.}
\begin{enumerate}[(a)]
\item Find the order of $(6,2,-k)$ in $\mathbb{Z}_{22} \times U(9) \times Q$ where $Q=\{\pm 1,\pm i, \pm j, \pm k\}$ is the quaternion group. 

\item Explain why $\mathbb{Z}_{20} \cong \mathbb{Z}_4 \times \mathbb{Z}_{5}$ but $\mathbb{Z}_{20} \not\cong \mathbb{Z}_{2} \times \mathbb{Z}_{10}$. 

         In addition, list the distinct orders of elements in both $\mathbb{Z}_4 \times \mathbb{Z}_{5}$ and $\mathbb{Z}_{2} \times \mathbb{Z}_{10}$.
         Give an example of an element of each such order.
\end{enumerate}

\item[\#3] Let $H$ and $K$ be normal subgroups of $G$. Show that $HK=\{hk \;|\; h \in H \text{ and } k \in K \}$ is a normal subgroup of $G$. 

You need to run through the normal subgroup test.

{\it Note:} It may be helpful to note that $HK=KH$. Why? Given $x \in HK$, we have $x=hk$ for some $h \in H$ and $k \in K$. Therefore, $x=hk \in Hk$. But $H$ is a normal subgroup of $G$ so that $Hk=kH$. Therefore, $x \in kH$ so that there exists some $h' \in H$ such that $x=kh'$. In other words, $x\in KH$. This shows that $HK \subseteq KH$. The proof of the other containment is similar. For a slicker proof: $HK = \bigcup\limits_{k\in K} Hk = \bigcup\limits_{k \in K} kH = KH$ (where the middle step used that $H$ is normal and thus its left and right cosets match).


\item[\#4] Let $H = \{ 1,x^3,y,x^3y \} \subseteq D_{6} = \{ 1,x,\dots,x^5,y,xy,\dots,x^5y \} = \langle x,y \;|\; x^6=1, y^2=1, xyxy=1 \rangle$. Make a table showing $H$ is closed under $D_6$'s operation (thus is a subgroup by the finite subgroup test). 

What is $[D_6:H]$ (i.e. the index of  $H$ in $D_6$)? 

Find all of the left and right cosets of $H$ in $D_6$. Is $H$ a normal subgroup of $D_6$?

\item[\#5] Let $G$ and $H$ be groups.
   \begin{enumerate}[(a)]
   \item Show $\{e\} \times H = \{ (e,h) \;|\; h \in H \}$ is a normal subgroup of  $G \times H$ (where $e$ is the identity of $G$).
   
   {\it Note:} You need to show that $\{e\} \times H$ is a subgroup AND that it's normal.
   
   \item Show $G \times H \cong H \times G$.
   \end{enumerate}
 
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