$\newcommand{\identity}{\mathrm{id}} \newcommand{\notdivide}{{\not{\mid}}} \newcommand{\notsubset}{\not\subset} \newcommand{\lcm}{\operatorname{lcm}} \newcommand{\gf}{\operatorname{GF}} \newcommand{\inn}{\operatorname{Inn}} \newcommand{\aut}{\operatorname{Aut}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\cis}{\operatorname{cis}} \newcommand{\chr}{\operatorname{char}} \newcommand{\Null}{\operatorname{Null}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&}$

# AppendixANotation

The following table defines the notation used in this book. Page numbers or references refer to the first appearance of each symbol.

Symbol Description Location
$a \in A$ $a$ is in the set $A$ Paragraph
${\mathbb N}$ the natural numbers Paragraph
${\mathbb Z}$ the integers Paragraph
${\mathbb Q}$ the rational numbers Paragraph
${\mathbb R}$ the real numbers Paragraph
${\mathbb C}$ the complex numbers Paragraph
$A \subset B$ $A$ is a subset of $B$ Paragraph
$\emptyset$ the empty set Paragraph
$A \cup B$ the union of sets $A$ and $B$ Paragraph
$A \cap B$ the intersection of sets $A$ and $B$ Paragraph
$A'$ complement of the set $A$ Paragraph
$A \setminus B$ difference between sets $A$ and $B$ Paragraph
$A \times B$ Cartesian product of sets $A$ and $B$ Paragraph
$A^n$ $A \times \cdots \times A$ ($n$ times) Paragraph
$id$ identity mapping Paragraph
$f^{-1}$ inverse of the function $f$ Paragraph
$a \equiv b \pmod{n}$ $a$ is congruent to $b$ modulo $n$ Example 1.2.30
$n!$ $n$ factorial Example 2.1.4
$\binom{n}{k}$ binomial coefficient $n!/(k!(n-k)!)$ Example 2.1.4
$a \mid b$ $a$ divides $b$ Paragraph
$\gcd(a, b)$ greatest common divisor of $a$ and $b$ Paragraph
$\mathcal P(X)$ power set of $X$ Exercise 2.3.12
$\lcm(m,n)$ the least common multiple of $m$ and $n$ Exercise 2.3.23
$\mathbb Z_n$ the integers modulo $n$ Paragraph
$U(n)$ group of units in $\mathbb Z_n$ Example 3.2.4
$\mathbb M_n(\mathbb R)$ the $n \times n$ matrices with entries in $\mathbb R$ Example 3.2.7
$\det A$ the determinant of $A$ Example 3.2.7
$GL_n(\mathbb R)$ the general linear group Example 3.2.7
$Q_8$ the group of quaternions Example 3.2.8
$\mathbb C^*$ the multiplicative group of complex numbers Example 3.2.9
$|G|$ the order of a group Paragraph
$\mathbb R^*$ the multiplicative group of real numbers Example 3.3.1
$\mathbb Q^*$ the multiplicative group of rational numbers Example 3.3.1
$SL_n(\mathbb R)$ the special linear group Example 3.3.3
$Z(G)$ the center of a group Exercise 3.4.48
$\langle a \rangle$ cyclic group generated by $a$ Theorem 4.1.3
$|a|$ the order of an element $a$ Paragraph
$\cis \theta$ $\cos \theta + i \sin \theta$ Paragraph
$\mathbb T$ the circle group Paragraph