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\parbox{1.95in}{\bf Math 3110}
\hfill {\Large \bf Homework \#7} \hfill
\parbox{1.95in}{\bf \hfill Due: {\bf Mon., Oct. 28${}^{\text{\bf th}}$\!, 2024}}

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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] Suppose $H$ and $K$ are subgroups of some group $G$ such that $|H|=55$ and $|K|=60$. 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 $(2, 12, j)$ in $U(15) \times \mathbb{Z}_{88} \times Q$ where $Q=\{\pm 1,\pm i, \pm j, \pm k\}$ is the quaternion group. 

\item Explain why $\mathbb{Z}_{99} \cong \mathbb{Z}_{11} \times \mathbb{Z}_{9}$ but $\mathbb{Z}_{99} \not\cong \mathbb{Z}_{3} \times \mathbb{Z}_{33}$. 

         In addition, list the distinct orders of elements in both $\mathbb{Z}_{11} \times \mathbb{Z}_{9}$ and $\mathbb{Z}_{3} \times \mathbb{Z}_{33}$.
         Give an example of an element of each such order.
\end{enumerate}

\item[\#3] 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[\#4] Let $G$ and $H$ be groups. Show $G \times \{e\} = \{ (g,e) \;|\; g \in G \}$ is a normal subgroup of  $G \times H$ (where $e$ is the identity of $H$).
   
   {\it Note:} You need to show that $G \times \{e\}$ is a subgroup AND that it's normal.

\item[\#5] Let $D=\{(g,g) \;|\; g \in G\}$ where $G$ is some group. 

{\it Note:} $D$ is the {\it diagonal} subgroup. It's all of the elements in $G\times G$ whose coordinates are equal.
 
\begin{enumerate}[(a)]
\item Show $D$ is a subgroup of $G \times G$. 

{\it ?Fun? Fact:} $D$ is not a normal subgroup of $G$ when $G$ is not abelian. Suppose $a,b \in G$ and $ab \not= ba$. Then $aba^{-1} \not= b$. Notice $(b,b) \in D$, but $(a,b)(b,b)(a,b)^{-1} = (a,b)(b,b)(a^{-1},b^{-1}) = (aba^{-1},bbb^{-1})=(aba^{-1},b) \not\in D$ since $aba^{-1}\not=b$.

\item Show $D \cong G$.
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