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 |
| \({\mathbb N}\) | 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 |