## ReferencesNotation

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