AATA Notation

Appendix C Notation

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
$N$	the natural numbers	Paragraph
$Z$	the integers	Paragraph
$Q$	the rational numbers	Paragraph
$R$	the real numbers	Paragraph
$C$	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
$A ∖ B$	difference between sets $A$ and $B$	Paragraph
$A \times B$	Cartesian product of sets $A$ and $B$	Paragraph
$A^{n}$	$A \times \dots \times A$ ( $n$ times)	Paragraph
$i d$	identity mapping	Paragraph
$f^{- 1}$	inverse of the function $f$	Paragraph
$a \equiv b (\mod n)$	$a$ is congruent to $b$ modulo $n$	Example 1.30
$n!$	$n$ factorial	Example 2.4
$(\binom{n}{k})$	binomial coefficient $n! / (k! (n - k)!)$	Example 2.4
$a ∣ b$	$a$ divides $b$	Paragraph
$gcd (a, b)$	greatest common divisor of $a$ and $b$	Paragraph
$P (X)$	power set of $X$	Exercise 2.4.12
$lcm (m, n)$	the least common multiple of $m$ and $n$	Exercise 2.4.23
$Z_{n}$	the integers modulo $n$	Paragraph
$U (n)$	group of units in $Z_{n}$	Example 3.11
$M_{n} (R)$	the $n \times n$ matrices with entries in $R$	Example 3.14
$det A$	the determinant of $A$	Example 3.14
$G L_{n} (R)$	the general linear group	Example 3.14
$Q_{8}$	the group of quaternions	Example 3.15
$C^{*}$	the multiplicative group of complex numbers	Example 3.16
$\| G \|$	the order of a group	Paragraph
$R^{*}$	the multiplicative group of real numbers	Example 3.24
$Q^{*}$	the multiplicative group of rational numbers	Example 3.24
$S L_{n} (R)$	the special linear group	Example 3.26
$Z (G)$	the center of a group	Exercise 3.5.48
$⟨ a ⟩$	cyclic group generated by $a$	Theorem 4.3
$\| a \|$	the order of an element $a$	Paragraph
$cis θ$	$\cos θ + i \sin θ$	Paragraph
$T$	the circle group	Paragraph
$S_{n}$	the symmetric group on $n$ letters	Paragraph
$(a_{1}, a_{2}, \dots, a_{k})$	cycle of length $k$	Paragraph
$A_{n}$	the alternating group on $n$ letters	Paragraph
$D_{n}$	the dihedral group	Paragraph
$[G : H]$	index of a subgroup $H$ in a group $G$	Paragraph
$L_{H}$	the set of left cosets of a subgroup $H$ in a group $G$	Theorem 6.8
$R_{H}$	the set of right cosets of a subgroup $H$ in a group $G$	Theorem 6.8
$a ∤ b$	$a$ does not divide $b$	Theorem 6.19
$d (x, y)$	Hamming distance between $x$ and $y$	Paragraph
$d_{min}$	the minimum distance of a code	Paragraph
$w (x)$	the weight of $x$	Paragraph
$M_{m \times n} (Z_{2})$	the set of $m \times n$ matrices with entries in $Z_{2}$	Paragraph
$Null (H)$	null space of a matrix $H$	Paragraph
$δ_{i j}$	Kronecker delta	Lemma 8.27
$G ≅ H$	$G$ is isomorphic to a group $H$	Paragraph
$Aut (G)$	automorphism group of a group $G$	Exercise 9.4.37
$i_{g}$	$i_{g} (x) = g x g^{- 1}$	Exercise 9.4.41
$Inn (G)$	inner automorphism group of a group $G$	Exercise 9.4.41
$ρ_{g}$	right regular representation	Exercise 9.4.44
$G / N$	factor group of $G$ mod $N$	Paragraph
$G^{'}$	commutator subgroup of $G$	Exercise 10.4.14
$\ker ϕ$	kernel of $ϕ$	Paragraph
$(a_{i j})$	matrix	Paragraph
$O (n)$	orthogonal group	Paragraph
$‖ x ‖$	length of a vector $x$	Paragraph
$S O (n)$	special orthogonal group	Paragraph
$E (n)$	Euclidean group	Paragraph
$O_{x}$	orbit of $x$	Paragraph
$X_{g}$	fixed point set of $g$	Paragraph
$G_{x}$	isotropy subgroup of $x$	Paragraph
$N (H)$	normalizer of s subgroup $H$	Paragraph
$H$	the ring of quaternions	Example 16.7
$Z [i]$	the Gaussian integers	Example 16.12
$char R$	characteristic of a ring $R$	Paragraph
$Z_{(p)}$	ring of integers localized at $p$	Exercise 16.7.33
$\deg f (x)$	degree of a polynomial	Paragraph
$R [x]$	ring of polynomials over a ring $R$	Paragraph
$R [x_{1}, x_{2}, \dots, x_{n}]$	ring of polynomials in $n$ indeterminants	Paragraph
$ϕ_{α}$	evaluation homomorphism at $α$	Theorem 17.5
$Q (x)$	field of rational functions over $Q$	Example 18.5
$ν (a)$	Euclidean valuation of $a$	Paragraph
$F (x)$	field of rational functions in $x$	Item 18.4.7.a
$F (x_{1}, \dots, x_{n})$	field of rational functions in $x_{1}, \dots, x_{n}$	Item 18.4.7.b
$a ⪯ b$	$a$ is less than $b$	Paragraph
$a \lor b$	join of $a$ and $b$	Paragraph
$a \land b$	meet of $a$ and $b$	Paragraph
$I$	largest element in a lattice	Paragraph
$O$	smallest element in a lattice	Paragraph
$a^{'}$	complement of $a$ in a lattice	Paragraph
$\dim V$	dimension of a vector space $V$	Paragraph
$U \oplus V$	direct sum of vector spaces $U$ and $V$	Item 20.5.17.b
$Hom (V, W)$	set of all linear transformations from $U$ into $V$	Item 20.5.18.a
$V^{*}$	dual of a vector space $V$	Item 20.5.18.b
$F (α_{1}, \dots, α_{n})$	smallest field containing $F$ and $α_{1}, \dots, α_{n}$	Paragraph
$[E : F]$	dimension of a field extension of $E$ over $F$	Paragraph
$GF (p^{n})$	Galois field of order $p^{n}$	Paragraph
$F^{*}$	multiplicative group of a field $F$	Paragraph
$G (E / F)$	Galois group of $E$ over $F$	Paragraph
$F_{{σ_{i}}}$	field fixed by the automorphism $σ_{i}$	Proposition 23.14
$F_{G}$	field fixed by the automorphism group $G$	Corollary 23.15
$Δ^{2}$	discriminant of a polynomial	Exercise 23.5.22

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