References Notation
Symbol | Description | Location |
---|---|---|
\(\C\) | the complex numbers | Paragraph |
\(\R\) | the real numbers | Paragraph |
\(|z|\) | norm of the complex number \(|z|\) | Paragraph |
\(\arg(z)\) | argument of the complex number \(z\) | Paragraph |
\(z^\ast\) | conjugate of the complex number \(z\) | Paragraph |
\(\Quat\) | the quaternions | Paragraph |
\(\R^3_\Quat\) | the space of pure quaternions | Paragraph |
\(U(\Quat)\) | the unit quaternions | Paragraph |
\(S^1\) | unit circle in the plane | Paragraph |
\(\extR\) | extended real numbers | Paragraph |
\(S^2\) | unit sphere in \(\R^3\) | Paragraph |
\(\extC\) | extended complex numbers | Paragraph |
\([x]\) | the equivalence class of an element \(x\) | Paragraph |
\(X/\!\!\sim\) | the set of equivalence classes for an equivalence relation \(\sim\) | Paragraph |
\(\Perm(X)\) | permutations of a set \(X\) | Definition 2.1.1 |
\(S_n\) | the symmetric group on \(n\) symbols | Definition 2.1.1 |
\(R_\theta\) | planar rotation by angle \(\theta\) | Assemblage |
\(F_L\) | planar reflection across line \(L\) | Assemblage |
\(D_n\) | dihedral group | Definition 2.1.5 |
\(C_n\) | the \(n\)th roots of unity | Paragraph |
\(GL(n,\R)\) | the group of \(n\times n\) invertible matrices with real entries | Definition 2.1.13 |
\(GL(n,\C)\) | the group of \(n\times n\) invertible matrices with complex entries | Definition 2.1.13 |
\(\F^\ast\) | group of nonzero elements in a field \(\F\) | Definition 2.1.14 |
\(|G|\) | order of the group \(G\) | Definition 2.2.5 |
\(C(a)\) | the centralizer of an element \(a\) in a group \(G\) | Exercise 2.3.2.5 |
\(Z(G)\) | the center of a group \(G\) | Exercise 2.3.2.5 |
\(G\approx H\) | group \(G\) is isomorphic to group \(H\) | Definition 2.4.1 |
\( H\trianglelefteq G \) | \(H\) is a normal subgroup of \(G\) | Definition 2.4.8 |
\(\Aut(G)\) | the group of automorphisms of a group \(G\) | Exercise 2.4.2.14 |
\(\Inn(G)\) | group of inner automorphisms of a group \(G\) | Item 2.4.2.14.b |
\(\Orb(x)\) | orbit of \(x\) under a group action | Definition 2.5.1 |
\(\Stab(x)\) | stabilizer of an element \(x\) under a group action | Definition 2.5.1 |
\(X/G\) | set of orbits of the action of group \(G\) on set \(X\) | Proposition 2.5.4 |
\(\Proj(V) \) | projective space | Exercise 2.5.3.6 |
\(PGL(V)\) | the projective linear group | Item 2.5.3.6.d |
\(U_n\) | group of units in \(\Z_n\) | Exercise 2.6.1 |
\(A_n\) | the alternating group | Item 2.6.3.b |
\(F \cong F'\) | figure \(F\) is congruent to figure \(F'\) | Definition 3.1.1 |
\(\M\) | the Möbius transformation group | Definition 3.2.4 |
\(\D\) | the open unit disk | Definition 3.3.1 |
\(\H\) | the hyperbolic group | Definition 3.3.1 |
\(\U\) | the upper half-plane | Definition 3.3.19 |
\(\HU\) | the upper half-plane group | Definition 3.3.19 |
\(SU(2)\) | the special unitary group | Paragraph |
\(R_{v,\theta}\) | rotation about the axis \(v\) by angle \(\theta\) | Paragraph |
\(\S\) | the elliptic group | Paragraph |