📚 The CoCalc Library - books, templates and other resources
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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 2.3.12 |
| $\lcm(m,n)$ | the least common multiple of $m$ and $n$ | Exercise 2.3.23 |
| $\mathbb Z_n$ | 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 3.4.48 |
| $\langle a \rangle$ | 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 9.3.37 |
| $i_g$ | $i_g(x) = gxg^{-1}$ | Exercise 9.3.41 |
| $\inn(G)$ | inner automorphism group of a group $G$ | Exercise 9.3.41 |
| $\rho_g$ | right regular representation | Exercise 9.3.44 |
| $G/N$ | factor group of $G$ mod $N$ | Paragraph |
| $G'$ | commutator subgroup of $G$ | Exercise 10.3.14 |
| $\ker \phi$ | 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 16.6.34 |
| $\deg f(x)$ | 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 18.3.7.a |
| $F(x_1, \dots, x_n)$ | field of rational functions in $x_1, \ldots, x_n$ | Item 18.3.7.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 20.4.17.b |
| $\Hom(V, W)$ | set of all linear transformations from $U$ into $V$ | Item 20.4.18.a |
| $V^*$ | dual of a vector space $V$ | Item 20.4.18.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 23.13 |
| $F_G$ | field fixed by the automorphism group $G$ | Corollary 23.14 |
| $\Delta^2$ | discriminant of a polynomial | Exercise 23.4.22 |