appendix-3.ipynb

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AppendixCNotation

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 1.30
$n!$	$n$ factorial	Example 2.4
$\binom{n}{k}$	binomial coefficient $n!/(k!(n-k)!)$	Example 2.4
$a \mid b$	$a$ 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 3.11
$\mathbb M_n(\mathbb R)$	the $n \times n$ matrices with entries in $\mathbb R$	Example 3.14
$\det A$	the determinant of $A$	Example 3.14
$GL_n(\mathbb R)$	the general linear group	Example 3.14
$Q_8$	the group of quaternions	Example 3.15
$\mathbb C^*$	the multiplicative group of complex numbers	Example 3.16
$|G|$	the order of a group	Paragraph
$\mathbb R^*$	the multiplicative group of real numbers	Example 3.24
$\mathbb Q^*$	the multiplicative group of rational numbers	Example 3.24
$SL_n(\mathbb R)$	the special linear group	Example 3.26
$Z(G)$	the center of a group	Exercise 3.4.48
$\langle a \rangle$	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
$[G:H]$	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 6.8
$\mathcal R_H$	the set of right cosets of a subgroup $H$ in a group $G$	Theorem 6.8
$a \notdivide b$	$a$ does not divide $b$	Theorem 6.19
$d(\mathbf x, \mathbf y)$	Hamming distance between $\mathbf x$ and $\mathbf y$	Paragraph
$d_{\min}$	the minimum distance of a code	Paragraph
$w(\mathbf x)$	the weight of $\mathbf x$	Paragraph
$\mathbb M_{m \times n}(\mathbf Z_2)$	the set of $m \times n$ matrices with entries in $\mathbb Z_2$	Paragraph
$\Null(H)$	null space of a matrix $H$	Paragraph
$\delta_{ij}$	Kronecker delta	Lemma 8.27
$G \cong H$	$G$ is isomorphic to a group $H$	Paragraph
$\aut(G)$	automorphism group of a group $G$	Exercise 9.3.37
$i_g$	$i_g(x) = gxg^{-1}$	Exercise 9.3.41
$\inn(G)$	inner automorphism group of a group $G$	Exercise 9.3.41
$\rho_g$	right regular representation	Exercise 9.3.44
$G/N$	factor group of $G$ mod $N$	Paragraph
$G'$	commutator subgroup of $G$	Exercise 10.3.14
$\ker \phi$	kernel of $\phi$	Paragraph
$(a_{ij})$	matrix	Paragraph
$O(n)$	orthogonal group	Paragraph
$\| {\mathbf x} \|$	length of a vector $\mathbf x$	Paragraph
$SO(n)$	special orthogonal group	Paragraph
$E(n)$	Euclidean group	Paragraph
${\mathcal O}_x$	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
$\mathbb H$	the ring of quaternions	Example 16.7
$\mathbb Z[i]$	the Gaussian integers	Example 16.12
$\chr R$	characteristic of a ring $R$	Paragraph
$\mathbb Z_{(p)}$	ring of integers localized at $p$	Exercise 16.6.34
$\deg f(x)$	degree of a polynomial	Paragraph
$R[x]$	ring of polynomials over a ring $R$	Paragraph
$R[x_1, x_2, \ldots, x_n]$	ring of polynomials in $n$ indeterminants	Paragraph
$\phi_\alpha$	evaluation homomorphism at $\alpha$	Theorem 17.5
$\mathbb Q(x)$	field of rational functions over $\mathbb Q$	Example 18.5
$\nu(a)$	Euclidean valuation of $a$	Paragraph
$F(x)$	field of rational functions in $x$	Item 18.3.7.a
$F(x_1, \dots, x_n)$	field of rational functions in $x_1, \ldots, x_n$	Item 18.3.7.b
$a \preceq b$	$a$ is less than $b$	Paragraph
$a \vee b$	join of $a$ and $b$	Paragraph
$a \wedge b$	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 20.4.17.b
$\Hom(V, W)$	set of all linear transformations from $U$ into $V$	Item 20.4.18.a
$V^*$	dual of a vector space $V$	Item 20.4.18.b
$F( \alpha_1, \ldots, \alpha_n)$	smallest field containing $F$ and $\alpha_1, \ldots, \alpha_n$	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
$F_{\{\sigma_i \}}$	field fixed by the automorphism $\sigma_i$	Proposition 23.13
$F_G$	field fixed by the automorphism group $G$	Corollary 23.14
$\Delta^2$	discriminant of a polynomial	Exercise 23.4.22