\documentclass[11pt,twoside]{amsart}
\usepackage{amssymb, amsmath, enumerate, palatino, hyperref}

%%% Theorem environments %%%
\theoremstyle{remark}
\newtheorem{prob}{Problem}
%%%%%%%%%%%%%%%%%%

%%% Macros %%%%%%%%%%%
\newcommand{\CC}{\mathbb{C}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\ZZ}{\mathbb{Z}}
%%%%%%%%%%%%%%%%%%

\title{Math 332: Homework 1}
%\author{Your name here --- delete the % at the start of the line}
\date{2.II.15}

\begin{document}
\maketitle

\noindent Unless otherwise specified, $(G,\cdot)$ is a group.  We will often refer to $(G,\cdot)$ as simply $G$, and write $ab$ for $a\cdot b$ when $a,b\in G$.

\begin{prob}
For a positive integer $n$, let $\mu_n(\CC)$ denote the set of complex numbers whose $n$-th power is $1$.  Prove that $(\mu_n(\CC),\cdot)$ is a group where $\cdot$ is the usual multiplication of complex numbers.  (This group is frequently called the \emph{group of $n$-th roots of unity}.)  Also prove that $(\mu_n(\CC),+)$ is not a group where $+$ is the usual addition of complex numbers.
\end{prob}

\begin{prob}
For a positive integer $n$, let $\ZZ_n = \{0,1,\ldots,n-1\}$ and define an operation $\boxplus:\ZZ_n\times \ZZ_n\to \ZZ_n$ such that $a\boxplus b$ is the remainder of $a+b$ after dividing by $n$.  Show that $\boxplus$ is well-defined and that $(\ZZ_n,\boxplus)$ is a group.  (Later, we will write $(\ZZ/n\ZZ,+)$ for $(\ZZ_n,\boxplus)$ and call it the \emph{group of integers mod $n$}.)  Also prove that $(\ZZ_n,\boxtimes)$ is not a group where $a\boxtimes b$ is the remainder of $a\cdot b$ after dividing by $n$.
\end{prob}

\begin{prob}
Prove that $(\mu_n(\CC),\cdot)$ is isomorphic to $(\ZZ_n,\boxplus)$.  \emph{Bonus}: Invent other groups which are isomorphic to $\mu_n(\CC)$ and $\ZZ_n$.
\end{prob}

\begin{prob}
Let $x$ be an element of $G$.  Suppose $|x| = n$ for some positive integer $n$.  Prove that $x^{-1} = x^{n-1}$.
\end{prob}

\begin{prob}
For $x,y\in G$, prove that $xy=yx$ if and only if $y^{-1}xy = x$ if and only if $x^{-1}y^{-1}xy=1$.
\end{prob}

\begin{prob}
Compute the order of each of the elements of each of the following groups: $D_6$, $D_8$, and $D_{10}$.
\end{prob}

\begin{prob}
Let $T$ be the group of rigid motions (\emph{aka} rotations) of a regular tetrahedron.  Show that $|T| = 12$.  \emph{Bonus}: Let $C$, $O$, $I$, and $D$ be the groups of rigid motions of the regular cube, octahedron, icosahedron, and dodecahedron.  Find $|C|$, $|O|$, $|I|$, and $|D|$.
\end{prob}

\begin{prob}
Suppose $\sigma,\tau\in S_{15}$ have cycle decompositions
\[
\begin{aligned}
  \sigma &= (1\enspace 13\enspace 5\enspace 10)(3\enspace 15\enspace8)(4\enspace14\enspace11\enspace7\enspace12\enspace9)\\
  \tau &= (1\enspace14)(2\enspace9\enspace15\enspace13\enspace4)(3\enspace10)(5\enspace12\enspace7)(8\enspace11).
\end{aligned}
\]
Find the cycle decompositions of $\sigma^2$, $\sigma\tau$, $\tau\sigma$, and $\tau^2\sigma$.
\end{prob}

\begin{prob}
Let $\NN = \{0,1,2,\ldots\}$ denote the set of natural numbers.  Prove that $S_\NN$ is an infinite group.  \emph{Bonus}:  Can you say anything more precise about the cardinality of $S_\NN$?
\end{prob}

\begin{prob}
If $A$, $B$, and $C$ are groups, prove that
\[
  A\times B\cong B\times A\quad\text{and}\quad A\times(B\times C)\cong (A\times B)\times C.
\]
\end{prob}

\begin{prob}
Let $G$ be any group.  Prove that the map $G\to G$ taking $g\mapsto g^{-1}$ is a homomorphism if and only if $G$ is abelian.  What conditions guarantee that $g\mapsto g^{-1}$ is an automorphism?
\end{prob}

\begin{prob}
Prove that $D_8$ and $Q_8$ are not isomorphic.
\end{prob}

\begin{prob}
A \emph{subgroup} of a group $G$ is a subset $H\subseteq G$ such that
\begin{enumerate}[(1)]
\item $1\in H$,
\item if $a,b\in H$, then $ab\in H$, and
\item if $a\in H$, then $a^{-1}\in H$.
\end{enumerate}
Prove that $H$ is a group under the operation $\cdot$ restricted to $H$.
\end{prob}

\begin{prob}
Let $G$ and $H$ be groups and let $\varphi:G\to H$ be a homomorphism.  Define the \emph{kernel} of $\varphi$ to be
\[
  \ker \varphi = \{g\in G \mid \varphi(g) = 1\}.
\]
Prove that $\ker \varphi$ is a subgroup of $G$.  Prove that $\varphi$ is injective if and only if $\ker \varphi = \{1\}$.
\end{prob}

\begin{prob}
Recall that $Q_8$ is the quaternion group of order $8$ with generators $i$, $j$.  Prove that the map $\varphi$ from $Q_8$ to $GL_2(\CC)$ defined on generators by
\[
  \varphi(i) = \begin{pmatrix} \sqrt{-1} &0\\ 0&-\sqrt{-1}\end{pmatrix}\quad\text{and}\quad \varphi(j) = \begin{pmatrix} 0& -1\\1& 0\end{pmatrix}
\]
extends to a homomorphism.  Prove that $\varphi$ is in fact a monomorphism.
\end{prob}

\end{document}