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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. 24${}^{\text{\bf th}}$\!, 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] Let $H$ and $K$ be subgroups of $G$.
   \begin{enumerate}[(a)]
   \item Suppose that $H$ and $K$ are normal subgroups of $G$. Show that $H \cap K$ is a normal subgroup of $G$ as well.
   
   {\it Note:} Please include a careful proof that $H \cap K$ is a subgroup -- even though we've shown this before.
   
   \item Let $|G|=100$, $|H|=50$, and $|K|=20$. Using Lagrange's Theorem, what are the possible orders of $H \cap K$?
   \end{enumerate}
\item[\#2] Let $H = \{ 1,x^4,y,x^4y \} \subseteq D_{8} = \{ 1,x,\dots,x^7,y,xy,\dots,x^7y \} = \langle x,y \;|\; x^8=1, y^2=1, xyxy=1 \rangle$. It isn't hard to show that $H$ is closed under the operation in $D_8$, thus by the finite subgroup test, $H$ is a subgroup of $D_8$.

Quickly compute $[D_8:H]$ (i.e. the index of  $H$ in $D_8$). Then find all of the left and right cosets of $H$ in $D_8$. Is $H$ a normal subgroup of $D_8$?

%\item Let $G = \langle a \rangle$ be some cyclic group and suppose that $|a|=15$. Find all of the left cosets of $H=\langle a^5 \rangle$ in $G = \langle a \rangle$.
\item[\#3] Direct products of cyclic groups.\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 $(5,6,44)$ in $\mathbb{Z}_9 \times \mathbb{Z}_{8} \times \mathbb{Z}_{66}$. 
\item Explain why $\mathbb{Z}_{50} \cong \mathbb{Z}_2 \times \mathbb{Z}_{25}$ but $\mathbb{Z}_{50} \not\cong \mathbb{Z}_{5} \times \mathbb{Z}_{10}$. 

         In addition, list the distinct orders of elements in both $\mathbb{Z}_2 \times \mathbb{Z}_{25}$ and $\mathbb{Z}_{5} \times \mathbb{Z}_{10}$ and give an example of an element of each such order.
\end{enumerate}
\item[\#4] 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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