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Introduction to Groups and Geometries
David W. Lyons
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\(\DeclareMathOperator{\sgn}{sgn} \DeclareMathOperator{\Inn}{Inn} \DeclareMathOperator{\lcm}{lcm} \DeclareMathOperator{\Aut}{Aut} \DeclareMathOperator{\Perm}{Perm} \DeclareMathOperator{\Stab}{Stab} \DeclareMathOperator{\Orb}{Orb} \DeclareMathOperator{\Rot}{Rot} \DeclareMathOperator{\re}{Re} \DeclareMathOperator{\im}{Im} \DeclareMathOperator{\img}{image} \DeclareMathOperator{\conj}{conj} \DeclareMathOperator{\Id}{Id} \def\expi{E} \def\wrap{W} \newcommand{\C}{\mathbb{C}} \newcommand{\Quat}{\mathbb{H}} \newcommand{\extC}{\hat{\C}} \newcommand{\R}{\mathbb{R}} \newcommand{\extR}{\hat{\R}} \newcommand{\F}{\mathbb{F}} \newcommand{\extF}{\hat{\F}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Proj}{\mathbb{P}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\M}{{\rm MOB}} \newcommand{\E}{{\rm EUC}} \renewcommand{\H}{{\rm HYP}} \newcommand{\HU}{\rm HYP_{\U}} \renewcommand{\S}{{\rm ELL}} \newcommand{\D}{\mathbb{D}} \newcommand{\closedD}{\hat{\D}} \newcommand{\U}{\mathbb{U}} \newcommand{\spacer}{\rule[0cm]{0cm}{0cm}} \newcommand{\MOD}{\mathbin{\text{MOD}}} \newcommand{\twotwo}[4]{\left[ \begin{array}{cc} #1 \amp #2 \\ #3 \amp #4 \end{array} \right]} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \definecolor{fillinmathshade}{gray}{0.9} \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} \)
Front Matter
Colophon
Preface
About the author
1
Preliminaries
1.1
The Complex Plane
1.1
Exercises
1.2
Quaternions
1.2.1
Cartesian form and pure quaternions
1.2.2
Correspondence with complex matrices
1.2.3
Addition and multiplication
1.2.4
Conjugate, modulus, and polar form
1.2.5
Quaternions as rotations of
\(\R^3_\Quat\)
1.2.6
Exercises
1.3
Stereographic projection
1.3.1
Stereographic projection
\(S^1\to \extR\)
1.3.2
Stereographic projection
\(S^2\to \extC\)
1.3.3
Conjugate Transformations
1.3.4
Exercises
1.4
Equivalence relations
1.4.1
Definitions
1.4.2
Important example: the integers modulo an integer
\(n\)
1.4.3
Facts
1.4.4
Exercises
1.5
More preliminary topics
1.5.1
A useful tool: commutative diagrams
1.5.2
Exercises
2
Groups
2.1
Examples of groups
2.1.1
Permutations
2.1.2
Symmetries of regular polygons
2.1.3
The norm 1 complex numbers
2.1.4
The
\(n\)
-th roots of unity
2.1.5
Integers
2.1.6
Invertible matrices
2.1.7
Nonzero elements in a field
2.1.8
Unit quaternions
2.1.9
Exercises
2.2
Definition of a group
2.2
Exercises
2.3
Subgroups and cosets
2.3
Exercises
2.4
Group homomorphisms
2.4
Exercises
2.5
Group actions
2.5
Exercises
2.6
Additional exercises
2.6
Exercises
3
Geometries
3.1
Geometries and models
3.1.1
Examples of geometries
3.1.2
Planar geometries
3.1.3
Subgeometries and equivalent geometries
3.1.4
Exercises
3.2
Möbius geometry
3.2.1
Möbius transformations
3.2.2
The Fundamental Theorem of Möbius Geometry
3.2.3
Cross ratio
3.2.4
Clines (generalized circles)
3.2.5
Symmetry with respect to a cline
3.2.6
Normal forms
3.2.7
Steiner circles
3.2.8
Exercises
3.3
Hyperbolic geometry
3.3.1
The hyperbolic transformation group
3.3.2
Classification of clines in hyperbolic geometry
3.3.3
Normal forms for the hyperbolic group
3.3.4
Hyperbolic length and area
3.3.5
The upper-half plane model
3.3.6
Exercises
3.4
Elliptic geometry
3.4.1
The group of unit quaternions
3.4.2
The group of rotations of the 2-sphere
3.4.3
The elliptic subgroup of the Möbius group
3.4.4
Circles in
\(S^2\)
and clines in
\(\extC\)
3.4.5
Angles and orientation on
\(S^2\)
3.4.6
Elliptic length and area
3.4.7
Exercises
3.5
Projective geometry
3.5.1
Projective points, lines, and flats
3.5.2
Coordinates
3.5.3
Freedom in projective transformations
3.5.4
The real projective plane
3.5.5
Exercises
3.6
Additional exercises
3.6
Exercises
Back Matter
Further topics
References
Index
Notation
Introduction to Groups and Geometries
David W. Lyons
Department of Mathematical Sciences
Lebanon Valley College
Annville, PA, USA
lyons@lvc.edu
May 2023 Edition, revised: November 25, 2024
Colophon
Preface
About the author
