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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