## Index Index

Abelian group, Definition

action of a group on a set, Definition

\(G\)-space, Definition

faithful group action, Definition

isotropy subgroup, Definition

orbit of an element under a group action, Definition

Orbit-Stabilizer Theorem, Theorem

stabilizer of an element under a group action, Definition

transitive group action, Definition

additive notation, Definition

alternating group, Item

argument of a complex number, Paragraph

associative binary operation, Paragraph

asymptotic \(n\)-gon in hyperbolic space, Exercise

automorphism group, Exercise

Cartesian form of a complex number, Paragraph

Cartesian form of a quaternion, Paragraph

center (of a group), Exercise

centralizer of a group element, Exercise

circle group, Definition

circles of Apollonius, Paragraph

cline, Paragraph

commutative binary operation, Definition

commutative diagram, Paragraph

complex number, Subsection

conformal transformation, Paragraph

congruence transformations, Definition

congruent, Definition

conjugate of a complex number, Paragraph

conjugate of a quaternion, Paragraph

conjugate transformations, Exercises

conjugation by a group element, Exercise

coset, Definition

cosets of a normal subgroup, Proposition

cross ratio of Möbius geometry, Paragraph

cross ratio of projective geometry, Exercise

cyclic group, Exercise

cyclic permutations, Exercise

digraph, Paragraph

directed graph, Paragraph

path in a directed graph, Paragraph

disjoint cyclic permutations, Exercise

doubly-asymptotic hyperbolic triangle, Exercise

elliptic geometry

2-gon, Exercise

Möbius subgeometry model, Definition

quaternion model, Definition

spherical model, Definition

elliptic group, Paragraph

elliptic Möbius transformation, Paragraph

equivalence class, Paragraph

equivalence relation, Paragraph

equivalent geometries, Definition

Euclidean planar geometry, Item

Euler phi function, Exercise

Euler's formula, Assemblage

extended complex numbers, Paragraph

extended real numbers, Paragraph

factor group, Proposition

faithful group action, Definition

Fermat's Little Theorem, Exercise

figure, Definition

First Isomorphism Theorem, Corollary

full cycle, Exercise

Fundamental Theorem of Möbius geometry, Proposition

Fundamental Theorem of Projective Geometry, Theorem

general linear group, Definition

generalized circle, Paragraph

generators of a (sub)group, Proposition

geometries, list of

elliptic geometry

Möbius subgeometry model, Definition

quaternion model, Definition

spherical model, Definition

Euclidean planar geometry, Item

hyperbolic geometry

disk model, Definition

upper half-plane model, Definition

Möbius geometry, Definition

projective geometry, Item

spherical geometry, Item

geometry, Definition

great circle, Paragraph

group, Definition

group action, Definition

group automorphism, Exercise

group of units in \(\Z_n\), Exercise

groups, list of

alternating group, Item

circle group, Definition

dihedral group, Definition

elliptic group, Paragraph

general linear group, Definition

group of permutations of a set, Definition

group of units in \(\Z_n\), Exercise

Möbius group, Definition

projective linear group, Item

special unitary group, Paragraph

symmetric group, Definition

the \(n\)-th roots of unity, Definition

the unit quaternions, Paragraph

trivial group, Definition

upper half-plane transformation group, Definition

homomorphism of groups, Definition

hyperbolic distance, Paragraph

hyperbolic geometry

disk model, Definition

upper half-plane model, Definition

hyperbolic group, Definition

hyperbolic Möbius transformation, Paragraph

hyperbolic rotation, Paragraph

hyperbolic translation, Paragraph

identity element, Definition

imaginary number, Paragraph

imaginary part of a complex number, Paragraph

inner automorphism, Exercise

inner automorphism group, Item

inverse of a group element, Definition

isomorphism of groups, Definition

kernel of a group homomorphism, Definition

Lagrange's Theorem, Assemblage

left multiplication, Exercise

linear fractional transformation, Paragraph

loxodromic Möbius transformation, Paragraph

model space, Definition

modulus of a complex number, Paragraph

modulus of a quaternion, Paragraph

multiplicative notation, Definition

Möbius geometry, Definition

Möbius group, Definition

Möbius transformation, Paragraph

one-step subgroup test, Proposition

Orbit-Stabilizer Theorem, Theorem

order of a group, Definition

order of a group element, Exercise

Pappus' Theorem, Assemblage

parabolic Möbius transformation, Paragraph

parallel displacement, Paragraph

parity of a permutation, Exercise

partition of a set, Paragraph

permutation

\(k\)-cycle, Exercise

full cycle, Exercise

permutation of a set, Paragraph

permutation group of a set, Definition

Poincaré disk model of hyperbolic geometry, Definition

polar form of a complex number, Paragraph

polar form of a quaternion, Paragraph

product group, Exercise

projective geometry, Item

\(k\)-dimensional flat, Paragraph

conic, Exercise

coordinate chart, Exercise

cross ratio, Exercise

homogeneous coordinates, Paragraph

inhomogeneous coordinates, Paragraph

line in projective space, Paragraph

projective line, Exercise

quadric, Exercise

real projective plane, Paragraph

projective linear group, Item

projective space, Exercise

Pythagorean Theorem

Elliptic Pythagorean Theorem, Item

Hyperbolic Pythagorean Theorem, Item

quaternion

imaginary part of a quaternion, Paragraph

imaginary quaternion, Paragraph

pure quaternion, Paragraph

real part of a quaternion, Paragraph

real quaternion, Paragraph

scalar part of a quaternion, Paragraph

scalar quaternion, Paragraph

unit quaternions, Paragraph

vector part of a quaternion, Paragraph

quotient group, Proposition

real part of a complex number, Paragraph

rectangular form of a complex number, Paragraph

reflection (in the plane), Assemblage

rotation (in the plane), Assemblage

semidirect product of groups, Exercise

special unitary group, Paragraph

spherical geometry, Item

Steiner circles, Paragraph

degenerate Steiner circles, Paragraph

Steiner circles of the first kind, Paragraph

Steiner circles of the second kind, Paragraph

stereographic projection, Subsection

subgeometry, Definition

subgroup, Definition

subgroup generated by a set, Proposition

symmetric group, Definition

the dihedral group, Definition

transitive group action, Definition

two-step subgroup test, Proposition

upper half-plane model of hyperbolic geometry, Definition

well-defined function, Paragraph