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References Notation

Symbol Description Location
\(\Quat\) 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
\(\Z\) the integers 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 2.3.5
\(Z(G)\) the center of a group \(G\) Exercise 2.3.5
\(G\approx H\) 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 2.4.15
\(\Inn(G)\) group of inner automorphisms of a group \(G\) Item 2.4.15.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 \(X\) Definition 2.5.4
\(A_n\) the alternating group Item 2.5.1.h
\(\Proj(V)\) projective space Exercise 2.5.7
\(PGL(V)\) the projective linear group Item 2.5.7.d
\(U_n\) group of units in \(\Z_n\) Exercise 2.6.1
\(F \cong F'\) figure \(F\) is congruent to figure \(F'\) Definition 3.1.1
\(\M\) Möbius transformation group Definition 3.2.3
\(\D\) the open unit disk Definition 3.3.1
\(\H\) the hyperbolic group Definition 3.3.1
\(\U\) the upper half-plane Definition 3.3.24
\(\HU\) the upper half-plane group Definition 3.3.24
\(SU(2)\) the special unitary group Paragraph
\(R_{v,\theta}\) rotation about the axis \(v\) by angle \(\theta\) Paragraph
\(\S\) the elliptic group Paragraph