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\noindent
\parbox{2.25in}{\bf Math 3110-101}
\hfill {\Large \bf Homework \#5} \hfill
\parbox{2.25in}{\bf \hfill Due: Fri., Mar. $27^\mathrm{th}$, 2015}

\vspace{0.2in}

\begin{enumerate}
\item Let $H$ and $K$ be subgroups of $G$.
   \begin{enumerate}[(a)]
   \item Prove that $H \cap K$ is a subgroup of $G$.
   \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 $|G|=100$, $|H|=50$, and $|K|=20$. Using Lagrange's Theorem, what are the possible orders of $H \cap K$?
   \end{enumerate}
\item Let $H = \{ 1,x^2,x^4 \} \subseteq D_6 = \{ 1,x,\dots,x^5,y,xy,\dots,x^5y \}$. Notice $H = \langle x^2 \rangle$, so $H$ is a subgroup of $D_6$.
          Quickly compute $[D_6:H]$ (i.e. the index of  $H$ in $D_6$). Then find all of the left and right cosets of $H$ in $D_6$. Is $H$ a normal subgroup of $D_6$?
\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 Explain why $\mathbb{Z}_{12} \cong \mathbb{Z}_3 \oplus \mathbb{Z}_4$ but $\mathbb{Z}_{12} \not\cong \mathbb{Z}_2 \oplus \mathbb{Z}_6$.
\item Let $G$ and $H$ be groups.
   \begin{enumerate}[(a)]
   \item Show $G \oplus H$ is abelian if and only if $G$ and $H$ are abelian.
   \item Show $G \oplus H \cong H \oplus G$.
   \end{enumerate}
\end{enumerate}



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