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# AppendixBNotation

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 5.30
$n!$ $n$ factorial Example 6.34
$\binom{n}{k}$ binomial coefficient $n!/(k!(n-k)!)$ Example 6.34
$a \mid b$ $a$ divides $b$ Paragraph
$\gcd(a, b)$ greatest common divisor of $a$ and $b$ Paragraph
$\mathbb Z_n$ the integers modulo $n$ Paragraph
$U(n)$ group of units in $\mathbb Z_n$ Example 2.11
$\mathbb M_n(\mathbb R)$ the $n \times n$ matrices with entries in $\mathbb R$ Example 2.14
$\det A$ the determinant of $A$ Example 2.14
$GL_n(\mathbb R)$ the general linear group Example 2.14
$Q_8$ the group of quaternions Example 2.15
$\mathbb C^*$ the multiplicative group of complex numbers Example 2.16
$|G|$ the order of a group Paragraph
$\mathbb R^*$ the multiplicative group of real numbers Example 3.9
$\mathbb Q^*$ the multiplicative group of rational numbers Example 3.9
$SL_n(\mathbb R)$ the special linear group Example 3.11
$Z(G)$ the center of a group Exercise 3.4.15
$\langle a \rangle$ cyclic group generated by $a$ Theorem 4.3
$|a|$ the order of an element $a$ Paragraph
$\cis \theta$ $\cos \theta + i \sin \theta$ Paragraph
$\mathbb T$ the circle group Paragraph
$S_n$ the symmetric group on $n$ letters Paragraph
$(a_1, a_2, \ldots, a_k )$ cycle of length $k$ Paragraph
$A_n$ the alternating group on $n$ letters Paragraph
$D_n$ the dihedral group Paragraph
$G \cong H$ $G$ is isomorphic to a group $H$ Paragraph
$\aut(G)$ automorphism group of a group $G$ Exercise 6.4.37
$i_g$ $i_g(x) = gxg^{-1}$ Exercise 6.4.41
$\inn(G)$ inner automorphism group of a group $G$ Exercise 6.4.41
$\rho_g$ right regular representation Exercise 6.4.44
$[G:H]$ index of a subgroup $H$ in a group $G$ Paragraph
$\mathcal L_H$ the set of left cosets of a subgroup $H$ in a group $G$ Theorem 7.8
$\mathcal R_H$ the set of right cosets of a subgroup $H$ in a group $G$ Theorem 7.8
$a \notdivide b$ $a$ does not divide $b$ Theorem 7.19
$G/N$ factor group of $G$ mod $N$ Paragraph
$G'$ commutator subgroup of $G$ Exercise 9.4.14
$\ker \phi$ kernel of $\phi$ Paragraph