Index Index
Abelian group, Definition
action of a group on a set
isotropy subgroup, Definition
Orbit-Stabilizer Theorem, Theorem
transitive group action, Definition
additive notation, Definition
alternating group, Item
associative binary operation, Paragraph
asymptotic \(n\)-gon in hyperbolic space, Exercise
automorphism group, Exercise
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
conformal transformation, Paragraph
congruence transformations, Definition
congruent, Definition
conjugate of a quaternion, Paragraph
conjugate transformations, Paragraph
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
cycle notation, Exercise
cyclic group, Exercise
cyclic permutation, 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
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
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
\(G\)-space, Definition
faithful group action, Definition
orbit of an element under a group action, Definition
orbit space, Definition
stabilizer of an element under a 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
homothety, Table
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
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 quaternion, Paragraph
multiplicative notation, Definition
Möbius geometry, Definition
Möbius group, Definition
Möbius transformation, Paragraph
normal form, Paragraph
normal subgroup, Definition
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
partition of a set, Paragraph
permutation
\(k\)-cycle, Exercise
cycle notation, Exercise
cyclic permutation, Exercise
even permutation, Exercise
odd permutation, Exercise
permutation of a set, Paragraph
sign function, Exercise
transposition, Exercise
permutation group of a set, Definition
Poincaré disk model of hyperbolic geometry, Definition
polar form of a quaternion, Paragraph
product group, Exercise
projective geometry, Item
\(k\)-dimensional flat, Paragraph
collinear projective points, 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
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
transposition, Exercise
two-step subgroup test, Proposition
upper half-plane model of hyperbolic geometry, Definition
well-defined function, Paragraph
