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\noindent
\parbox{2.25in}{\bf Math 3110-101}
\hfill {\Large \bf Homework \#6} \hfill
\parbox{2.25in}{\bf \hfill Due: Mon., Oct. $26^\mathrm{th}$, 2015}

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\begin{enumerate}
\item 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.
   \item Let $|G|=36$, $|H|=12$, and $|K|=18$. Using Lagrange's Theorem, what are the possible orders of $H \cap K$?
   \end{enumerate}
\item Let $H = \{ 1,x^3,x^6 \} \subseteq D_9 = \{ 1,x,\dots,x^8,y,xy,\dots,x^8y \} = \langle x,y \;|\; x^9=1, y^2=1, xyxy=1 \rangle$. Notice $H = \langle x^3 \rangle$, so $H$ is a subgroup of $D_9$.  Quickly compute $[D_9:H]$ (i.e. the index of  $H$ in $D_9$). Then find all of the left and right cosets of $H$ in $D_9$. Is $H$ a normal subgroup of $D_9$?
%\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 Direct products of cyclic groups.
\begin{enumerate}[(a)]
\item Find the order of $(5,2,6)$ in $\mathbb{Z}_6 \oplus \mathbb{Z}_{8} \oplus \mathbb{Z}_{99}$. 
\item Explain why $\mathbb{Z}_{20} \cong \mathbb{Z}_4 \oplus \mathbb{Z}_5$ but $\mathbb{Z}_{20} \not\cong \mathbb{Z}_2 \oplus \mathbb{Z}_{10}$.
\end{enumerate}
\item Let $G$ and $H$ be groups.
   \begin{enumerate}[(a)]
   \item Show $\{e\} \oplus H = \{ (e,h) \;|\; h \in H \}$ is a normal subgroup of  $G \oplus H$ (where $e$ is the identity of $G$).\\
   {\it Note:} You need to show that $\{e\} \oplus H$ is a subgroup AND that it's normal.
   \item Show $G \oplus H \cong H \oplus G$.
   \end{enumerate}
\end{enumerate}



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