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\begin{document}
\title{Homework $\#$3}
\author{Due 10/9/13 at start of class }
\maketitle
\subsection*{Problem 1 (D$\&$F 2.1.1,2.1.2)} In each of the following, determine (and prove) whether or not the specified subset is a subgroup of the given group.
\begin{itemize}
\item[(a)] The set of 2-cycles in $S_n$ for $n\geq 3$.
\item[(b)] For fixed $n\in\mathbb{Z}^+$, the set of rational numbers whose denominators divide $n$ (under addition).
\item[(c)] The set of complex numbers of the form $a+ai$, $a\in\mathbb{R}$ (under addition).
\item[(d)] The set of odd integers in $\mathbb{Z}$ together with $0$ (under addition).
\item[(e)] The set of reflections in $D_{2n}$ for $n\geq 3$.
\end{itemize}  

\subsection*{Problem 2 (D$\&$F 1.6.13,14)} Let $G$ and $H$ be groups and let $\phi:G\rightarrow H$ be a homomorphism. Recall that the \textit{kernel} of $\phi$ is the subset $\{g\in G\;\vert\; \phi(G)=e_{H}\}\subseteq G$, while the \textit{image} of $\phi$ is the subset $\{\phi(g)\;\vert\; g\in G\} \subseteq H$. Prove that the kernel of $\phi$ is a subgroup of $G$ and the image of $\phi$ is a subgroup of $H$. 


\subsection*{Problem 3 (D$\&$F 2.1.5)} Let $G$ be a group with $\vert G\vert =n>2$. Prove that $G$ cannot have a subgroup $H$ with $\vert H\vert =n-1$. Do not use Lagrange's Theorem.

\subsection*{Problem 4 (D$\&$F 2.1.8)} Let $H$ and $K$ be subgroups of $G$. Prove that $H\cup K$ is a subgroup of $G$ if and only if either $H\subseteq K$ or $K\subseteq H$.

\subsection*{Challenge Problem} Let $H$ be a subgroup of the additive group of rational numbers with the property that $\frac{1}{x}\in H$ for every nonzero element $x$ of $H$. Prove that either $H=0$ or $H=\mathbb{Q}$.

\subsection*{Problem 5} Find all subgroups of $Z_{32}=\langle x\rangle$, giving a generator for each. Describe the containments between these subgroups.  

\subsection*{Problem 6 (D$\&$F 2.3.12)} Prove that the following groups are \textit{not} cyclic:
\begin{itemize}
\item[(a)] $Z_2\times Z_2$
\item[(b)] $Z_2\times\mathbb{Z}$
\item[(c)] $\mathbb{Z}\times\mathbb{Z}$
\end{itemize}

\subsection*{Problem 7 } Let $G$ be a group with an element $g\neq e$ such that $g$ is an element of every subgroup of $G$, except the trivial subgroup $\{e\}$. Prove that every element of $G$ has finite order.

\subsection*{Problem 8} Show that $Z_n$ has an element as described in Problem $8$ if and only if $n$ is a power of a prime.
\end{document}