## 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 1.34
$\binom{n}{k}$ binomial coefficient $n!/(k!(n-k)!)$ Example 1.34
$\mathcal P(X)$ power set of $X$ Exercise 1.5.37
$\mathbb Z_n$ 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 2.24
$\mathbb Q^*$ the multiplicative group of rational numbers Example 2.24
$SL_n(\mathbb R)$ the special linear group Example 2.26
$\langle a \rangle$ cyclic group generated by $a$ Theorem 2.34
$|a|$ the order of an element $a$ 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
$Z(G)$ the center of a group Exercise 2.7.42
$[G:H]$ 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 3.8
$\mathcal R_H$ the set of right cosets of a subgroup $H$ in a group $G$ Theorem 3.8
$G/N$ factor group of $G$ mod $N$ Paragraph
$G'$ commutator subgroup of $G$ Exercise 3.3.31
$G \cong H$ $G$ is isomorphic to a group $H$ Paragraph
$\ker \phi$ kernel of $\phi$ Paragraph
$\aut(G)$ automorphism group of a group $G$ Exercise 4.4.37
$i_g$ $i_g(x) = gxg^{-1}$ Exercise 4.4.41
$\inn(G)$ inner automorphism group of a group $G$ Exercise 4.4.41
$\rho_g$ right regular representation Exercise 4.4.44
$\mathbb H$ the ring of quaternions Example 5.7
$\mathbb Z[i]$ the Gaussian integers Example 5.12
$\chr R$ characteristic of a ring $R$ Paragraph
$\mathbb Z_{(p)}$ ring of integers localized at $p$ Exercise 5.6.33
$a \mid b$ $a$ divides $b$ Paragraph
$\gcd(a, b)$ greatest common divisor of $a$ and $b$ Paragraph
$\lcm(m,n)$ the least common multiple of $m$ and $n$ Exercise 6.1.5.8
$\deg f(x)$ 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 6.12
$\cis \theta$ $\cos \theta + i \sin \theta$ Paragraph
$\mathbb T$ the circle group Paragraphs
$\mathbb Q(x)$ field of rational functions over $\mathbb Q$ Example 7.5
$\nu(a)$ Euclidean valuation of $a$ Paragraph
$F(x)$ field of rational functions in $x$ Item 7.4.7.a
$F(x_1, \dots, x_n)$ field of rational functions in $x_1, \ldots, x_n$ Item 7.4.7.b
$F( \alpha_1, \ldots, \alpha_n)$ smallest field containing $F$ and $\alpha_1, \ldots, \alpha_n$ Paragraph
$\dim V$ dimension of a vector space $V$ Paragraph
$[E:F]$ dimension of a field extension of $E$ over $F$ Paragraph
$G(E/F)$ Galois group of $E$ over $F$ Paragraph
$F_{\{\sigma_i \}}$ field fixed by the automorphism $\sigma_i$ Proposition 9.19
$F_G$ field fixed by the automorphism group $G$ Corollary 9.20
$G \cong H$ $G$ is isomorphic to a group $H$ 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
$\langle a \rangle$ cyclic group generated by $a$ Theorem 11.3
$|a|$ the order of an element $a$ Paragraph
$a \notdivide b$ $a$ does not divide $b$ Theorem 11.26
${\mathcal O}_x$ 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
$(a_{ij})$ 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
$d(\mathbf x, \mathbf y)$ 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 16.27
$\gf(p^n)$ Galois field of order $p^n$ Paragraph
$F^*$ multiplicative group of a field $F$ Paragraph
$a \preceq b$ $a$ 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