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\title{Math 120: Homework 3}
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\begin{itemize}[noitemsep]
\item Section 1.7 \# 11, 17, 19;
\item Section 2.2 \# 7, 10;
\item Section 2.3 \# 17, 25;
\item Section 3.1 \# 9, 32, 33;
\end{itemize}

\bigbreak\noindent
{\textbf{Problem 1.7 \#11}}. Write out the cycle decomposition of the eight
permutations in $S_4$ corresponding to the elements of $D_8$ given by the
action of $D_8$ on the vertices of a square.

\bigbreak\noindent
{\textbf{Problem 1.7 \#17.}} Let $G$ be a group and let $G$ act on itself
by conjugation so that each $g \in G$ maps $G$ to $G$ by
\[ x \mapsto g x g^{- 1} . \]
For fixed $g \in G$ prove that conjugation by $g$ is an isomorphism of $G$ to
$G$ (i.e. an automorphism of $G$). Deduce that $x$ and $g x g^{- 1}$ have the
same order for all $x \in G$ and that for any subset $A$ of $G$, $| A | = | g
A g^{- 1} |$.

\bigbreak\noindent
{\textbf{Problem 1.7 \# 19.}} Let $H$ be a subgroup of the finite group $G$
and let $H$ act on $G$ by left multiplication. Let $x \in G$ and let
$\mathcal{O}$ be the orbit of $x$ under the action of $H$. Prove that the map
$H \mapsto \mathcal{O}$ defined by $h \mapsto h x$ is a bijection. From this
and the previous exercise deduce {\textit{Lagrange's theorem: if $G$ is a finite
group and $H$ is a subgroup then $| H |$ divides $| G |$}}.

\bigbreak\noindent
{\textbf{Problem 2.2 \# 7.}} Let $n \geqslant 3$. Prove
\smallskip\noindent
(a) $Z (D_{2 n}) = 1$ if $n$ is odd;
\smallskip\noindent
(b) $Z (D_{2 n}) = \{ 1, r^k \}$ of $n = 2 k$.

\bigbreak\noindent
{\textbf{Problem 2.2 \# 10.}} Let $H$ be a subgroup of order $2$ in $G$.
Show that $N_G (H) = C_G (H)$. Deduce that if $N_G (H) = G$ then $H \subseteq
Z (G)$.

\bigbreak\noindent
{\textbf{Problem 2.3 \# 17.}} Find a presentation of $Z_n$ with one
generator.

\bigbreak\noindent
{\textbf{Problem 2.3 \# 25.}} Let $G$ be a cyclic group of order $n$ and
let $k$ be an integer relatively prime to $n$. Prove the map $x \mapsto x^k$
is surjective. Use Lagrange's Theorem to prove the same is true for any finite
group of order $n$.

\bigbreak\noindent
{\textbf{Problem 3.1 \# 9.}} Define $\varphi : \mathbb{C}^{\times}
\longrightarrow \mathbb{R}^{\times}$ by $\varphi (a + b i) = a^2 + b^2$.
Prove that $\varphi$ is a homomorphism and find the image of $\varphi$.
Describe the kernel and fibers of $\varphi$ geometrically (as subsets of the
plane).

\bigbreak\noindent
{\textbf{Problem 3.1 \# 32.}} Prove that every subgroup of $Q_8$ is normal.
For each subgroup find the isomorphism type of its corresponding quotient.

\bigbreak\noindent
{\textbf{Problem 3.1 \# 33.}} Find all normal subgroups of $D_8$ and for
each of these find the isomorphism type of the corresponding quotient.

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