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
$x \in S$ $x$ is an element of $S$ Definition 1.1.3
$x \not\in S$ $x$ is not an element of $S$ Definition 1.1.3
$\emptyset$ the empty set, $\{\}$ Definition 1.1.3
$\Z$ the set of all integers Example 1.1.4
$\Q$ the set of all rational numbers Example 1.1.4
$\R$ the set of all real numbers Example 1.1.4
$\C$ the set of all complex numbers Example 1.1.4
$\N$ the set of all natural numbers, $\{0,1,2,\ldots\}$ Example 1.1.4
$\Z^+,\Q^+,\R^+$ the set of all positive elements of $\Z,\Q,\R$ Example 1.1.4
$\Z^-,\Q^-,\R^-$ the set of all negative elements of $\Z,\Q,\R$ Example 1.1.4
$\Z^*,\Q^*,\R^*,\C^*$ the set of all nonzero elements of $\Z,\Q,\R,\C$ Example 1.1.4
$\M_{m\times n}(S)$ the set of all $m \times n$ matrices over $S$ Definition 1.1.5
$\M_n(S)$ the set of all $n \times n$ matrices over $S$ Definition 1.1.5
$A\subseteq B$ $A$ is a subset of the $B$ Definition 1.1.8
$A\subsetneq B$ $A$ is a proper subset of $B$ Definition 1.1.8
$P(A)$ the power set of $A$ Definition 1.1.12
$A\cap B$ the intersection of $A$ and $B$ Definition 1.1.15
$A\cup B$ the union of $A$ and $B$ Definition 1.1.15
$A - B$ the difference of $A$ and $B$ Definition 1.1.15
$\bigcup_{i\in I}A_i$ $\{x: x\in A_i \text{ for some } i\in I\}$ Definition 1.1.15
$\bigcap_{i\in I}A_i$ $\{x: x\in A_i \text{ for every } i\in I\}$ Definition 1.1.15
$A\times B$ the direct product of $A$ and $B$ Definition 1.1.16
$f:S\to T$ function $f$ from $S$ to $T$ Definition 1.2.1
$f(U)$ the image of a set $U$ under $f$ Definition 1.2.1
$f^{\leftarrow}(V)$ the preimage of a set $V$ under $f$ Definition 1.2.1
$f\circ g$ the composition of $f$ with $g$ Definition 1.2.7
$1_S$ the identity function on $S$ Definition 1.2.7
$f^{-1}$ the inverse of $f$ Theorem 1.2.10
$|S|$ the cardinality of $S$ Definition 1.3.1
$\langle S, *\rangle$ binary structure Definition 2.1.1
$e$ the identity element in a binary structure/group Definition 2.1.6
$\det A$ the determinant of $A$ Definition 2.4.3
$GL(n,\R)$ the general linear group of degree $n$ over $\R$ Definition 2.4.3
$I_n$ the $n\times n$ identity matrix Theorem 2.4.4
$e_G$ the identity element in a group $G$ Convention 2.5.2
$a^{-1}$ the inverse of $a$ in a group Convention 2.5.2
$-a$ the inverse of $a$ in an abelian group Item
$n\Z$ $\{nm\,:\,m\in \Z\}$ Example 2.6.1
$a\equiv_n b$ $a$ is congruent to $b$ mod $n$ Definition 2.6.4
$R_n(a)$ the remainder when $a$ is divided by $n$ Definition 2.6.7
$+_n$ addition modulo $n$ Definition 2.6.8
$\Z_n$ the cyclic group of order $n$ Example 2.6.12
$\Z_n^{\times}$ $\{a\in \Z_n\,:\,\gcd(a,n)=1\}$ Definition 2.6.18
$F$ the set of all functions from $\R$ to $\R$ Example 2.6.20
$B$ the set of all bijections from $\R$ to $\R$ Example 2.6.21
$Z(G)$ the center of a group $G$ Exercise 2.8.9
$C^1$ the set of all differentiable functions from $\R$ to $\R$ whose derivatives are continuous Item 6
$C^0$ the set of all continuous functions from $\R$ to $\R$ Item 7
$c_a$ conjugation by $a$ Example 3.2.5
$G\simeq G'$ $G$ is isomorphic to $G'$ Definition 3.3.1
$G\not \simeq G'$ $G$ is not isomorphic to $G'$ Definition 3.3.1
$H\leq G$ $H$ is a subgroup of $G$ Definition 4.1.1
$H\not \leq G$ $H$ is not a subgroup of $G$ Definition 4.1.1
$\langle a \rangle$ the (cyclic) subgroup generated by $a$ Definition 5.1.5
$o(a)$ the order of element $a$ Definition 5.1.10
$S_A$ the set of all permutations on $A$ Definition 6.1.6
$S_n$ the symmetric group on $n$ letters Definition 6.2.1
$A_n$ the alternating group on $n$ letters Definition 6.3.8
$\lambda_a$ left multiplication by $a$ Definition 6.4.2
$\rho_a$ right multiplication by $a$ Definition 6.4.2
$\mapsto$ maps to Paragraph
$D_n$ the dihedral group of order $2n$ Definition 6.5.4
$xRy$ $x$ is related to $y$ Definition 7.1.3
$x\not R y$ $x$ is not related to $y$ Definition 7.1.3
$[x]$ the equivalence class of $x$ Definition 7.1.9
$a\siml b$ $a^{-1}b\in H\text{,}$ where $H\leq G$ is specified Definition 7.2.1
$a\simr b$ $ab^{-1}\in H\text{,}$ where $H\leq G$ is specified Definition 7.2.1
$aH, a+H$ the left coset of $H$ containing $a$ Definition 7.2.4
$Ha, H+a$ the right coset of $H$ containing $a$ Definition 7.2.4
$\Leftrightarrow$ if and only if Note 7.2.5
$H\unlhd G$ $H$ is a normal subgorup of $G$ Definition 7.2.13
$G/H$ the set of all left cosets of $H$ in $G$ Definition 7.2.18
$(G:H)$ $|G/H|$ Definition 7.3.1
$aHb$ $\{ahb\,:h\in H\}$ Definition 8.2.1
$\Ker \phi$ the kernel of $\phi$ Definition 8.2.7
$G/N$ the factor group $G/N\text{,}$ when $N\unlhd G$ Definition 8.3.2
$\Psi$ the canonical epimorphism from $G$ to $G/N$ Definition 8.3.14
$S^1$ the unit circle $\{e^{i\theta} \,:\, \theta\in \R\}$ in the complex plane Paragraph