Introduction

Fundamental knowldge for the following topics is assumed. You may want to check the appendix and my other notes for these.

• Linear Algebra
• Group Theory

Resources Used

• Groups and Symmetry by M. A. Armstrong
• Symmetries by D.L. Johnson
• Cut The Knot website
• From Groups to Geometry and Back by Vaughn Climenhaga and Anatole Katok

Notation

• 0 \in \N and \N^+ :=\ \N \setminus \{0\}.
• \varnothing denotes the empty set.

Status

These notes are still under work therefore not complete, also

• there are not many exercises,
• proofs might be absent,
• typos are quite possible,
• ordering is generally good but should be improved, and
• more visuals and intuition should be provided.

Todo's

• Delve into the complex plane?
  • Define conformality and show every isometry is conformal
• Explain (formally prove) group-theoric concepts with geometry
  • Especially better explain homomorphism, isomorphism and their groups and finally semidirect products
• Add figures drawn on excalidraw.
• Beckman–Quarles theorem
• Affine transformation
• Homotethy
• Shear Transform
• Spiral Similarity
• Inversion
  • Stereographic Projection and Inversion
  • Möbius Transformations

Def. Isometry

Let (X, d_X) and (Y, d_Y) be metric spaces. A map f: X \to Y is called an isometry or distance-preserving map if for all a, b \in X,

Two metric spaces (X, d_X) and (Y, d_Y) are called isometric if there is a bijective isometry between them.

Thm. Basic Properties of Isometries

Let f: X \to Y be an isometry on the metric spaces (X, d_X) and (Y, d_Y), then

1. f is always injective.
2. f is not necessarily surjective. For example x \mapsto (x,0) is an isometry but not surjective on \R \to \R^2.
3. Composition of isometries are an isometry.

Exercise

Thm. Inverse of an Isometry

Let f: X \to Y be an isometry. If f is bijective then f^{-1} is also an isometry. Consider the case f is not bijective, then we can construct an bijective isometry

so that since f is injective (and thus f') and f' is surjective we have, by construction, f' bijective.

Thm. Isometries are Continuous

An isometry f: X \to Y is always continuous.

Def. \text{Isom}(X)

Let (X, d) be a metric space, then set of all bijective isometries on X denoted \text{Isom}(X) forms a group under composition called the (full) isometry group of X.

Exercise.

Thm. \text{Isom}(\Z)

We have

where \bold{D}_\infty is the infinite dihedral group.

Exercises

Def. Symmetry

Let (X,d) be a metric space and S \subseteq X. A symmetry of S is an isometry f: X \to X such that

Def. \text{Sym}(S)

Let (X,d) be a metric space and S \subseteq X, then the set of all symmetries of S forms a group, and thus is a subgroup of \text{Isom}(X).

Exercise

Example. \text{Sym}(\Z)

The group of symmetries of \Z is generated by two elements, particularly by a translation and a reflection so that

where t(n) = n+1 and r(n) = -n.

Def. Euclidian Isometry

From now on we will call an isometry f: \R^n \to \R^n under the Euclidian distance an Euclidian isometry.

Notice that \tilde{f}(x) := f(x) - f(0), which we will call reduced Euclidian isometry of f, is an origin-fixing Euclidian isometry so that \tilde{f}(0) = 0.

Thm. Euclidian Isometries are Bijective

Let f: \R^n \to \R^n be any Euclidian isometry, then f is bijective.

Thm. \text{Sym}(\R^n) = \text{Isom}(\R^n)

Obvious as each Euclidian isometry is bijective.

Thm. Norm Preservation

Let \tilde{f} be an origin-fixing Euclidian isometry, then \tilde{f} preserves the norm so that, for all x \in \R^n

Exercise

Thm. Inner Product Preservation

Let \tilde{f} be an origin-fixing Euclidian isometry, then \tilde{f} preserves the inner product so that, for all x,y \in \R^n

Thm. Additivity

Let \tilde{f} be an origin-fixing Euclidian isometry, then \tilde{f} is additive so that, for all x,y \in \R^n

TODO: Add not so simple proof which make use of norm expension and preservation

Thm. Euclidian Isometry Linearity

An Euclidian isometry f: \R^n \to \R^n is linear if and only if it fixes the origin.

Def. Polygon, n-gon

There is no established definition for a polygon. Currently think of it as an geometric object with a closed surface connected by closed polygonal chain.

An n-gon is a polygon with n edges.

Def. Regular n-gon

For n \geq 3, a regular n-gon denoted by \bold{P}_n is a (closed) polygon with n sides of equal length joined together at equal angles.

Thm. Regular n-gon Symmetries

A regular n-gon has 2n different symmetries which are n rotational symmetries and n reflection symmetries.

Thm. Dihedral Group

The symmetries of the regular n-gon \bold{P}_n forms a group under composition called the dihedral group denoted by \bold{D}_{2n}.

Some authors prefer the notation \bold{D}_n or \bold{Dih}_n, however notice that in our case 2n denotes the number of symmetries and its order.

\bold{D}_{2n} is the semidirect product of C_2 acting on C_n via the automorphism \varphi_s(r) = r^{-1}, therefore:

The group representation of \bold{D}_{2n} is as follows:

where r denotes a rotation and s denotes a reflection.

Notice that, by srs=r^{-1} condition, we have by induction: sr^k = r^{-k}s.

Def. Convex

A set is said to be convex if any two points A and B are contained in the set, the entire line segment [AB] also lies within the set.

Def. Polyhedron

A polyhedron is an union of finitely many convex polygons (called faces), arranged in space so that the intersection of any two polygons is a shared vertex or edge or the empty set.

This is the definition by O'Rourke (1993). Definitions of polyhedron (although differs) are equivalent and agreed upon.

Def. Platonic Solid

A platonic solid is a convex regular polyhedron in three-dimensional space. Therefore, its faces are congruent regular n-gons and at each vertex the same number of polygons (called faces) meet.

Thm. Euler's Polyhedron Formula

Any three-dimensional convex polyhedrons surface has an Euler characteristic of \chi = 2 so that

For the proof, please check out various online resources that are both geometric and algebraic.

Thm. Platonic Solids

There are only 5 platonic solids called (regular) tetrahedron, cube, (regular) octahedron, (regular) dodacahedron and (regular) icosahedron.

Thm. Cube Symmetries

Let G be the symmetry group of the cube including the rotations and the reflections, then

1. A cube has 24 rotations in total that consists of
   1. 9 rotations through 3 face axes
      1. 90\degree
      2. 180\degree
      3. 270\degree
   2. 8 rotations through 4 vertex axes
      1. 120\degree
      2. 240\degree
   3. 6 rotations through 6 edges
   4. the identity
2. G \cong \bold{S}_4 \times \Z_2 where \bold{S}_4 consists of rotations and \Z_2 consists of \Set{1,d} where d is the central inversion.
3. Every rotation of the cube maps vertices to vertices, so each diagonal is mapped to another diagonal.

Thm. Rotations on Cube

1. A non-trivial rotation fixes exactly one axis.
2. Two non-trivial rotations commute if and only if
   • They are on the same axis, or
   • Axes are perpendicular and both rotations are 180\degree.
3. Two rotations commute if and only if they preserve each others axis, that is, sends the others axis to itself.

Def. General Special Linear Groups

Let \mathbb{F} be a field and recall that the set of all invertible n \times n matrices form a group with matrix multiplication with the identity I_n.

Moreover, each matrix A in this group determines an invertible linear transformation:

where (\cdot)^T denotes the transpose matrix.

The group of invertible n\times n is matrices called the general linear group denoted with \text{GL}_n(\mathbb{F}). Simply \text{GL}_n or \text{GL}(n) if the field is known. Unless otherwise stated, we will assume the field is the real numbers \R.

The subgroup of \text{GL}_n(\mathbb{F}) which is composed of n \times n matrices with the determinant equal to +1 is called the special linear group and denoted with \text{SL}_n(\mathbb{F}).

Note that we are using column vector notation, in the row-vector notation f_A would be defined as xA^T.

Def. Orthogonal Matrix

 See the appendix for the formal definition of orthonormality.

An orthogonal (or orthonormal) matrix is a real square matrix whose columns and rows are orthonormal vectors. Equivalently, a matrix Q is orthogonal if

An orthogonal matrix is called special if its determinant is 1. Moreover, note that the determinant of an orthogonal matrix is either +1 or -1 (exercise).

Def. Orthogonal Group

Noting the orthogonal n \times n matrices forms a group under matrix multiplication, we also define orthogonal group \text{O}_n and special orthogonal group \text{SO}_n where the latter is the subgroup of the former with elements of determinant value of +1.

We have seen so far that if A \in O_n, then f_A preserves the distance and the inner product (and thus preserves orthogonality).

Thm. \text{O}_2 and \text{SO}_2 on the Unit Circle

Let A \in \text{O}_2, then the columns of A are unit vectors and are orthogonal to one another. Noting these vectors lie in the unit circle we get the following two representations for A:

Let A has determinant +1, then we obtain

which is an element of \text{SO}_2 and represents anti-clockwise rotation. Notice how this implies each element of \text{SO}_2 has the form e^{i \theta}. Indeed, there is an isomorphism between the unit circle in the complex plane and \text{SO}_2.

Now, if A has determinant -1, then we obtain

which represents reflection in a line at angle \frac{\theta}{2} to the positive x-axis.

Therefore, each 2 \times 2 orthogonal matrix represents either a rotation of the plane about the origin (determinant +1), or a reflection in a straight line through the origin (determinant -1).

Similarly, in the three-dimensional case, \text{SO}_3 represents a rotation of \R^3 about an axis which passes through the origin. Conversely, every rotation of \R^3 which fixes the origin is represented by a matrix in \text{SO}_3. From now on, we shall refer \text{SO}_3 as the rotation group in three dimensions.

Thm. Finite Subgroups of \text{O}_2

A finite subgroup of \text{O}_2 is either cyclic or dihedral.

 Thm. Finite Subgroups of \text{SO}_3

A finite subgroup of \text{SO}_3 is isomorphic to either

• cyclic group,
• dihedral group,
• rotational symmetry group of one of the regular (platonic) solids

Thm. Rotations

In the plane every rotation commute. In \R^3, rotations usually does not commute and the only matrix commuting with all of \text{SO}_3 is the identity matrix. In \R^4, both \Set{I, -I} commutes with every other element in \text{SO}_4.

Recall the fact \det(\lambda A) = \lambda^n \det(A) where \lambda is a constant.

Exercises

Def. (Group) Action

Recall that, formally, an action of a group G on a set X is a homomorphism from G to the automorphism group of X, namely \text{Aut}(X). We may also say G acts on X.

From now on assume we are given an action of a group on a set X.

Thm. Same Orbit Conjugates

Points in the same orbit have conjugate stabilizers, so that for all x and y in the same orbit, say gx = y, then

Thm. Orbit-Stabilizer Theorem

Let G be a group and x \in G, then

In order to denote the orbit of x, the notations

• \bar{x},
• G(x)
• Gx

are also used. Similarly, for the stabilizer of x, the notations

• G_x

could be used as well.

Notice the direct divisibility results on finite groups this theorem implies.

Thm. Burnside's Lemma

The Burnside's Lemma which also known as The Counting Theorem asserts

where

• |X/G| denotes the number of orbits, and
• |X^g| denotes the number of elements in X fixed by g. This is also sometimes denoted with |\text{Fix}(g)|.

Notice X^g is not the stabilizer of x which are the elements in G fixed by g, but rather elements fixed in X by g.

Thm. Center of \bold{D}_{2n}

Thm. \text{Isom}(\R^n)

Let \text{Isom}(\R^n) denote the set of all isometries \R^n \to \R^n, then it forms a group under composition.

Thm. Plane Isometry Properties

An isometry u of \R^2

1. maps any triangle ABC to a congruent triangle
2. preserves angles, and
3. maps lines to lines

Exercise

Thm. \text{Isom}(\R^2) Determinacy

An isometry of \R^2 is determined by its effect on any three non-collinear points.

Def. Translations

A translation t in \R^2 is a map that moves every point of the plane through a fixed distance in a fixed direction so that

where \bold{a} = (a_1, a_2) is a constant.

Three key properties of translations are

• they are orientation preserving,
• have no fixed points unless \bold{a} = \bold{0}, and
• compose according to t(\bold{a})t(\bold{b}) = t(\bold{a+b}).

Def. Rotation

A rotation s is a map that moves every point of the plane through a fixed angle \theta about a fixed point O called the centre.

Three key properties of rotations are

• they are orientation preserving,
• have just one fixed point, and
• and compose according to s(O, \alpha)s(O, \beta) = s(O, \alpha + \beta).

We will denote a rotation through \theta about the point O with \rho_{O,\theta} and will omit O if it is the origin.

Def. Reflection

A reflection r is a map that moves every point of the plane to its mirror-image in a fixed line \ell called the axis..

Three key properties of reflections are

• they are orientation reversing,
• fix every point on the line \ell, and
• satisfy r(\ell)^2 = \bold{1}.

We will denote a reflection through a line \ell with \sigma_\ell.

Thm. Normal Form Theorem

Fix a point O and line \ell in \R^2 with O \in \ell, then any isometry u of \R^2 can be written uniquely in the form

where

• r is a reflection in \ell
• \epsilon = \pm 1
• s is a rotation around the point O, and
• t is a translation.

Def. Glide Reflection

Given a pair P, P' of distinct points on a line \ell in \R^2, the isometry

is called a glide reflection. Such isometries have no fixed points and orientation reversing.

Thm. Product of Three Reflections

The product of three reflections in \R^2 is either a reflection or a glide reflection.

Thm. Classification of Plane Isometries

In \R^2, an isometry is either

• identity (d),
• reflection (db)
• rotation (dp)
• translation (dd), or
• glide reflection (dq)

Thm. Characterisation of Plane Isometries

For the 4 isometry types we defined so far, we have the following characterization table

Thm. Isometries in the Complex Plane

Every isometry of the Euclidian plane \R^2 viewed as the complex line \Complex is of one of the forms

• f(z) = \alpha z + \beta, or
• f(z) = \alpha \bar{z} + \beta

where \alpha, \beta \in \Complex and |\alpha| = 1.

Moreover, one is orientation-preserving and the other one is orientation-reversing.

Thm. General Plane Isometry

Let u be any plane isometry, then there exists an 2 \times 2 orthogonal matrix \bold{M} and a vector v \in \R^2 such that

for all x \in \R^2 where \bold{v} = u(\bold{0}) and \bold{M} is uniquely determined. Converse also holds i.e. this defines an isometry.

Therefore, the pair (\bold{v}, \bold{M}) where \bold{v} \in \R^2 and \bold{M} is an orthogonal 2 \times 2 matrix determines an isometry and vice-versa.

In row-vector notation, we would have u(\bold{x}) = \bold{v} + \bold{x}\bold{M}^T where \bold{x} and \bold{v} are row-vectors.

Thm. Explicit Classification of Plane Isometries

Before we continue, let u be an plane isometry, and define

then, if u is a

• translation, then is of the form (\bold{v} , \bold{I}) for some translation vector \bold{v},
• rotation anti-clockwise through \theta about the origin, then is of the form (\bold{0}, \bold{A}_\theta),
• reflection in the line \ell passing through the origin, then is of the form (\bold{0}, \bold{B}_{2\theta}) where \theta is the clockwise angle between the line and the x-axis.
• rotation anti-clockwise through \theta about the point \bold{c}, then is of the form (\bold{c} - \bold{A}_\theta \bold{c}, \bold{A}_\theta)
• reflection in any line m, then is of the form (2\bold{a}, \bold{B}_{2\theta}) where \bold{a} is the vector from the origin to the line — where \theta defined as above,
• glide reflection with the line m for points \bold{a} and \bold{b}, then is of the form (2 \bold{a} + \bold{b}, \bold{B}_{2\theta}) — where \theta and \bold{a} defined as above.

Thm. 3-product of Reflections

Every plane isometry is a composition of at most 3 reflections.

Note how every reflection changes orientation.

Moreover, the composition of three reflections in the plane is always an orientation-preserving isometry, hence it is either

• a reflection, or
• glide reflection.

Thm. Two Parallel Lines

Let \ell \parallel m, then \sigma_\ell \sigma_m is a translation through twice the directed distance from l to m. Converse also holds.

Thm. Two Instersecting Lines

Let two lines \ell and m meet at C with the non-zero oriented angle from \ell to m being \theta / 2, then

Converse also holds so that every rotation is a product of two reflections in intersecting lines.

Exercises

Thm. General Euclidian Isometry

Let f: \R^n \to \R^n be any Euclidian isometry, then there exists an orthogonal matrix \bold{A} \in \R^{n \times n} and a vector b \in \R^n such that

for all x \in \R^n where b = f(0) and \bold{A} is uniquely determined.

Thm. \text{Isom}(\R)

Let f \in \text{Isom}(\R), then f is of the form

with \epsilon = \pm 1 and b \in \R. Moreover, if

• \epsilon = 1, then it is a translation, and
• \epsilon = -1, then it is a reflection

Direct result of the theorem General Euclidian Isometry proven above.

Thm. Euclidian Isometries Preserve the Midpoint

Let f: \R^n \to \R^n be an Euclidian isometry, then it preserves the midpoints so that

Exercise.

Note that this is not necessarily the case for other isometries nor other other norms on \R^n.

Def. Euclidian Group

The Euclidian Group is a subgroup of the group of affine transformations. Precisely, it is the subgroup that preserve the Euclidian distance between any two points.

The isometries of the Euclidian n-space \R^n form a group under function composition called the Euclidian group denoted by \mathbb{E}^n. Equivalently, it could be defined as the semidirect product of translational group \bold{T}(n) and the orthogonal group \bold{O}(n) so that

In the Euclidian group, isometries are often split into two classed based on how they affect the orientation, namely direct isometries and opposite isometries.

Direct isometries forms a subgroup in \mathbb{E}(n) denoted with \mathbb{E}^+(n) and their linear part has determinant +1.

Opposite isometries are not a subgroup on their own, but form the other half of \mathbb{E}^n.

Thm. \mathbb{E}^2

In \mathbb{E}^2, every direct isometry is a translation or a rotation, and every opposite isometry is a reflection or a glide reflection.

 A reflection in a line followed by a translation parallel to the same line is called a glide reflection.

Def. Inner Product Space

An inner product space is a vector space \mathcal{V} over the field \mathbb{F} together with an inner product

that satisfies, for all vectors x,y,z \in \mathcal{V} and scalars a,b \in \mathbb{F}

1. \Braket{x,y} = \overline{\Braket{y, x}} called symmetry axiom
2. \Braket{x + y, z} = \Braket{x,z} + \Braket{y,z} called additivity axiom
3. \Braket{ax, y} = a\Braket{x,y} called homogenity axiom
4. \Braket{x,x} \geq 0 and \Braket{x,x} = 0 \iff x = 0 called the positivity axiom

Thm. Cauchy-Schwarz Inequality

Let x,y be two vectors in the inner product space \mathcal{V}, then

where \Vert \cdot \Vert is the norm induced by the inner product.

Def. Kronecker Delta Function

Def. Orthonormal

Let \mathcal{V} be an inner product space. A set of vectors \Set{v_1, v_2, ...} \subseteq \mathcal{V} is called orthonormal if

Each vector is orthogonal to each other.

Thm. Basic Properties of Orthonal Matrices

Let \bold{O} be an orthogonal matrix, then

1. \bold{O} is invertible.
2. \bold{O}^T \bold{O} = \bold{I}
3. The determinant of \bold{O} is either 1 or -1.
4. If \bold{O} is special orthogonal matrix, then — as a linear transformation — it acts as a rotation around a fixed point if n \geq 2 in \R^n.

Def. Semidirect Product

Let G be a group with the identity e, H \leq G and N \trianglelefteq G such that N \cap H = \Set{e}. We say G is the semidirect product of N and H if G=NH denoted by G = N \rtimes H or G = H \ltimes N.

Notice how \rtimes points toward the normal group similar to \trianglelefteq.

This is not a good definition of the semidirect product but a simple one. Curious reader might want to check out other resources.

Thm.

Let G be a group with a (normal) subgroup N of index 2. If there exists g \in G \setminus N of order 2, then