## 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 2.4.12
$$\lcm(m,n)$$ the least common multiple of $$m$$ and $$n$$ Exercise 2.4.23
$$\mathbb Z_n$$ 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 3.5.48
$$\langle a \rangle$$ 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 9.4.37
$$i_g$$ $$i_g(x) = gxg^{-1}$$ Exercise 9.4.41
$$\inn(G)$$ inner automorphism group of a group $$G$$ Exercise 9.4.41
$$\rho_g$$ 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
$$\ker \phi$$ 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 16.7.33
$$\deg f(x)$$ degree of a polynomial Paragraph
$$R[x]$$ ring of polynomials over a ring $$R$$ Paragraph
$$R[x_1, x_2, \ldots, x_n]$$ 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 18.4.7.a
$$F(x_1, \dots, x_n)$$ field of rational functions in $$x_1, \ldots, x_n$$ Item 18.4.7.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 20.5.17.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
$$F( \alpha_1, \ldots, \alpha_n)$$ 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 23.14
$$F_G$$ field fixed by the automorphism group $$G$$ Corollary 23.15
$$\Delta^2$$ discriminant of a polynomial Exercise 23.5.22