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\title{Math 120: Homework 2}
\author{}
\date{}

\maketitle

\begin{itemize}
  \item Section 1.2 \# 9,
  \item Section 1.3 \# 2 ($\sigma,\tau,\sigma\tau,\tau\sigma$ only), 5,13,20,
  \item Section 1.4 \# 7,
  \item Section 1.5 \# 2,
  \item Section 1.6 \# 14.
\end{itemize}

\noindent
{\textbf{Problem 1.2 \# 9}}. Let $G$ be the group of rigid motions of the
tetrahedron. Show that $| G | = 12$.

\medbreak\noindent
{\textbf{Problem 1.3 \# 2.}} Let $\sigma$ be the permutation
\[ \begin{array}{lllll}
     1 \mapsto 13, & 2 \mapsto 2, & 3 \mapsto 15, & 4 \mapsto 14, & 5 \mapsto
     10,\\
     6 \mapsto 6, & 7 \mapsto 12, & 8 \mapsto 3, & 9 \mapsto 4, & 10 \mapsto
     1,\\
     11 \mapsto 7, & 12 \mapsto 9, & 13 \mapsto 5, & 14 \mapsto 11, & 15
     \mapsto 8.
   \end{array} \]
and let $\tau$ be the permutation
\[ \begin{array}{lllll}
     1 \mapsto 14, & 2 \mapsto 9, & 3 \mapsto 10, & 4 \mapsto 2, & 5 \mapsto
     12,\\
     6 \mapsto 6, & 7 \mapsto 5, & 8 \mapsto 11, & 9 \mapsto 15, & 10 \mapsto
     3,\\
     11 \mapsto 8, & 12 \mapsto 7, & 13 \mapsto 4, & 14 \mapsto 1, & 15
     \mapsto 13.
   \end{array} \]
Find the cycle decompositions of the following permutations:
\[ \sigma, \tau, \sigma \tau, \tau \sigma . \]

\medbreak\noindent
{\textbf{Problem 1.3 \# 5.}} Find the order of
$(1, 12, 8, 10, 4) (2, 13) (5, 11, 7) (6, 9)$.

\bigbreak\noindent
{\textbf{Problem 1.3 \# 13.}} Show that an element has order 2 in $S_n$ if
and only if its cycle decomposition is a product of commuting $2$-cycles.

\bigbreak\noindent
{\textbf{Problem 1.3 \# 20.}} Find a set of generators and relations for
$S_3$.

\bigbreak\noindent
{\textbf{Problem 1.4 \# 7.}} Let $p$ be a prime. Show that the order of
$\operatorname{GL}_2 (\mathbb{F}_p)$ is $p^4 - p^3 - p^2 + p$.

\medbreak\noindent
{\textbf{Problem 1.5 \# 2.}} Write the group tables for $S_3, D_8$ and
$Q_8$.

\bigbreak\noindent
{\color{red}
{\textbf{Problem 1.6 \# 1.}} {\textbf{Not Assigned!}} Let $\varphi : G \longrightarrow H$ be a
homomorphism.

\medbreak\noindent
(a) Prove that $\varphi (x^n) = \varphi (x)^n$ for all $n \in \mathbb{Z}_+$.

\medbreak\noindent
(b) Show that $\varphi (x^{- 1}) = \varphi (x)^{- 1}$ and deduce that $\varphi (x^n) =
\varphi (x)^n$ for all $n$.
}

\bigbreak\noindent
{\textbf{Problem 1.6 \# 14.}} Let $G$ and $H$ be groups and let $\varphi : G
\rightarrow H$ be a homomorphism. Define the {\textit{kernel}} of $\varphi$ to
be $\{ g \in G| \varphi (g) = 1_H \}$. Prove that the kernel of $\varphi$ is a
subgroup of $G$. Prove that $\varphi$ is injective if and only if the kernel
of $\varphi$ is the identity subgroup $\{ 1_G \}$ of $G$.

\end{document}
