# Introduction to Groups and Geometries

## ReferencesNotation

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