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\begin{document}

\begin{center}
	\Large{Math 31 -- Homework 6}
	
	\vspace{2mm}
	\large{Due Friday, August 9}
\end{center}

\paragraph{Note:} Any problem labeled as ``show'' or ``prove'' should be written up as a formal proof, using complete sentences to convey your ideas.

% EASY
\subsection*{Easier}

\paragraph{1.} We will see in class that the kernel of any homomorphism is a normal subgroup. Conversely, you will show that any normal subgroup is the kernel of 
some homomorphism. That is, let $G$ be a group with $N$ a normal subgroup of $G$, and define a function $\pi : G \to G/N$ by
\[
	\pi(g) = Ng
\]
for all $g \in G$. Prove that $\pi$ is a homomorphism, and that $\ker \pi = N$.

\paragraph{2.} Recall that $\R^\times$ is the group of nonzero real numbers (under multiplication), and let $N = \{-1, 1\}$. Show that $N$ is a normal subgroup of 
$\R^\times$, and that $\R^\times/N$ is isomorphic to the group of positive real numbers under multiplication. [\textbf{Hint:} Use the Fundamental Homomorphism 
Theorem.]

\paragraph{3.} [Saracino, \#13.1] Let $\varphi : \Z_8 \to \Z_4$ be given by
	\[
		\varphi(x) = [x]_4,
	\]
	i.e., the remainder of $x$ mod 4. Find $\ker \varphi$. To which familiar group is $\Z_8/\ker \varphi$ isomorphic?

\paragraph{4.} If $G$ is a group and $M \normal G$, $N \normal G$, prove that $M \cap N \normal G$. [You proved on an earlier assignment that $M \cap N$ is a 
subgroup of $G$, so you only need to prove that it is normal.]

\paragraph{5.} Classify all abelian groups of order 600 up to isomorphism.


% MEDIUM
\subsection*{Medium}
\paragraph{6.} [Saracino, \#11.17 and 11.18] Let $G$ be a group, and let $H \leq G$.
\begin{thmenum}
	\item If $G$ is abelian, prove that $G/H$ is abelian. [\textbf{Hint:} You may want to use a result from the last homework assignment.]
	\item Prove that if $G$ is cyclic, then $G/H$ is also cyclic.
\end{thmenum}

\paragraph{7.} Prove that if $G_1$ and $G_2$ are abelian groups, then $G_1 \times G_2$ is abelian.

\paragraph{8.} If $G_1$ and $G_2$ are groups, prove that $G_1 \times G_2 \cong G_2 \times G_1$.

\paragraph{9.} Let $G = \Z_4 \times \Z_6$, and let $H = \la (1,0) \ra$. Find all the right cosets of $H$ in $G$, and compute the quotient group $G/H$. (That is,
identify it with a more familiar group.)










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