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 |