## 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 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 |