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\begin{document}

\begin{center}
	\Large{Math 31 -- Homework 4}
	
	\vspace{2mm}
	\large{Due Wednesday, July 17}
\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.

\subsection*{Easier}

\paragraph{1.} Determine whether each of the following subsets is a subgroup of the given group. If not, state which of the subgroup axioms fails.
\begin{thmenum}
	\item The set of real numbers $\R$, viewed as a subset of the complex numbers $\C$ (under addition).
	\item The set $\pi \Q$ of rational multiples of $\pi$, as a subset of $\R$ (under addition).
	\item The set of $n \times n$ matrices with determinant 2, as a subset of $\gln(\R)$.
	\item The set $\{i, m_1, m_2, m_3\} \subset D_3$ of reflections of the equilateral triangle, along with the identity transformation.
\end{thmenum}

\paragraph{2.} We proved in class that every subgroup of a cyclic group is cyclic. The following statement is almost the converse of this:
	\begin{center}
		``Let $G$ be a group. If every \emph{proper} subgroup of $G$ is cyclic, then $G$ is cyclic.''
	\end{center}
Find a counterexample to the above statement.

\paragraph{3.} [Saracino, \#5.10] Prove that any subgroup of an abelian group is abelian.

\subsection*{Medium}
\paragraph{4.} [Saracino, \#5.14] Let $G$ be a group. If $H$ and $K$ are subgroups of $G$, show that $H \cap K$ is also a subgroup of $G$.

\paragraph{5.} Let $r$ and $s$ be positive integers, and define 
\[
	H = \left\{nr + ms : n, m \in \Z \right\}.
\]
\begin{thmenum}
	\item Show that $H$ is a subgroup of $\Z$.
	\item We saw in class that every subgroup of $\Z$ is cyclic. Therefore, $H = \la d \ra$ for some $d \in \Z$. What is this integer $d$? Prove that the $d$ you've found is 
	in fact a generator for $H$.
\end{thmenum}

\paragraph{6.} Let $X$ be a set, and recall that $S_X$ is the group consisting of the bijections from $S$ to itself, with the binary operation given by composition of functions. (If $X$ is finite, then $S_X$ is just the symmetric group on $n$ letters, where $X$ has $n$ elements.) Given $x_1 \in X$, define
\[
	H = \left\{ f \in S_X : f(x_1) = x_1 \right\}.
\]
Show that $H \leq S_X$.

\paragraph{7.} [Saracino, \#5.22] Let $G$ be a group. Define
	\[
		Z(G) = \left\{ a \in G : ax = xa \text{ for all } x \in G \right\}.
	\]
In other words, the elements of $Z(G)$ are exactly those which commute with \emph{every} element of $G$. Prove that $Z(G)$ is a subgroup of $G$, called the \textbf{center} of $G$.

\paragraph{8.} Show that if $H$ and $K$ are subgroups of an \emph{abelian} group $G$, then
	\[
		\left\{ hk : h \in H \text{ and } k \in K \right\}
	\]
is a subgroup of $G$.

\paragraph{9.} [Saracino, \#5.20] We will see in class that if $p$ is a prime number, then the cyclic group $\Z_p$ has no proper subgroups as a consequence of 
Lagrange's theorem. This problem will have you investigate a ``converse'' to this result.
\begin{thmenum}
	\item If $G$ is a finite group which has no proper subgroups (other than $\{e\}$), prove that $G$ must be cyclic.
	\item Extend the result of (a) by showing that if $G$ has no proper subgroups, then $G$ is not only cyclic, but 
		\[
			\abs{G} = p
		\]
		for some prime number $p$.
\end{thmenum}

\subsection*{Hard}
\paragraph{10.} [Saracino, \#5.25 and 5.26] Let $G$ be a group, and let $H$ be a subgroup of $G$. 
	\begin{thmenum}
		\item Let $a$ be some fixed element of $G$, and define
			\[
				aHa^{-1} = \{ aha^{-1} : h \in H\}.
			\]
			This set is called the \textbf{conjugate} of $H$ by $a$. Prove that $aHa^{-1}$ is a subgroup of $G$.
			
		\item Define the \textbf{normalizer} of $H$ in $G$ to be
			\[
				N(H) = \{a \in G : aHa^{-1} = H\}.
			\]
			Prove that $N(H)$ is a subgroup of $G$.
	\end{thmenum}












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