UBS Businessplan. Strategic Planning and Financing Basis. Sample

WallpaperGroups

Lance Drager

Introductionto Groups ofIsometries

Symmetries ofan EquilateralTriangle

Groups ofIsometries

A Little GroupTheory

What is aGroup?

GroupTerminology

Group Maps

Group Maps andSubgroups

Exercises

Conjugation inthe Group ofPlane Isometries

Geometric Transformations and WallpaperGroups

Groups of Isometries

Lance Drager

Department of Mathematics and StatisticsTexas Tech University

Lubbock, Texas

2010 Math Camp

WallpaperGroups

Lance Drager



Groups ofIsometries


What is aGroup?

GroupTerminology

Group Maps


Exercises


Outline

1 Introduction to Groups of IsometriesSymmetries of an Equilateral TriangleGroups of Isometries

2 A Little Group TheoryWhat is a Group?Group TerminologyGroup MapsGroup Maps and SubgroupsExercisesConjugation in the Group of Plane Isometries

WallpaperGroups

Lance Drager



Groups ofIsometries


What is aGroup?

GroupTerminology

Group Maps


Exercises


Symmetries of a Triangle

• Problem. Find all the symmetries of an equilateraltriangle.

A B

C

WallpaperGroups

Lance Drager



Groups ofIsometries


What is aGroup?

GroupTerminology

Group Maps


Exercises


All the Symmetries

• There are six, the identity, (call it e), rotation by 120◦,call it r , r2 and the reflections SA, SB and SC = s in themirror lines MA, MB and MC

A B

C

r

MAMB

MC

WallpaperGroups

Lance Drager



Groups ofIsometries


What is aGroup?

GroupTerminology

Group Maps


Exercises


Comparing Products 1

• Here’s SB .

A B

C A

BC

SB

• Here’s rs.

A B

C

AB

C

s

A

BC

r

WallpaperGroups

Lance Drager



Groups ofIsometries


What is aGroup?

GroupTerminology

Group Maps


Exercises


Comparing Products 2

• Here’s SA.

A B

C

A

B

C

SA

• Here’s r2s.

A B

C

AB

C

s

A

B

C

r2

WallpaperGroups

Lance Drager



Groups ofIsometries


What is aGroup?

GroupTerminology

Group Maps


Exercises


Find the Product

• What is sr?

A B

C

A

B

C

r

A

B

C

s

• Answer: SA = r2s

A B

C

A

B

C

SA

WallpaperGroups

Lance Drager



Groups ofIsometries


What is aGroup?

GroupTerminology

Group Maps


Exercises


Multiplication Table

• The set of symmetries is D3 = {e, r , r2, s, rs, r2s} and wehave the relations r3 = e, s2 = e and sr = r2s.

• Multiplication table: (Row label)× (Column label)

e r r2 s rs r2s

e e r r2 s rs r2sr r r2 e rs r2s sr2 r2 e r r2s s rss s r2s rs e r2 rrs rs s r2s r e r2

r2s r2s rs s r2 r e

WallpaperGroups

Lance Drager



Groups ofIsometries


What is aGroup?

GroupTerminology

Group Maps


Exercises


Relations and MultiplicationExamples

• The table is completely determined by the relationsr3 = e, s2 = e and sr = r2s.

• Examples from the Multiplication Table• Example: (r2s)(rs) = r2srs = r2(sr)s = r2(r2s)s = r4s2 =

r4e = rr3 = re = r .• Example: (rs)(r2s) = rsr2s = r(sr)rs = r(r2s)rs = r3srs =

srs = (sr)s = (r2s)s = r2s2 = r2.

WallpaperGroups

Lance Drager



Groups ofIsometries


What is aGroup?

GroupTerminology

Group Maps


Exercises


Groups of Isometries

• D3 is a group of isometries. In math, this means threethings

1 D3 contains the identity.2 D3 is closed under taking products3 D3 is closed under taking inverses.

• Dn is the group of symmetries of a regular n-gon. It isgenerated by a rotation r and a reflection s, with therelations rn = e and sr = rn−1s. It has 2n elements.

ExerciseVerify that the group of symmetries of a square is D4. If youhave the endurance, write down the multiplication table.Verify that the group of symmetries of the pentagon is D5.(Don’t write down the multiplication table, unless you insist.)

WallpaperGroups

Lance Drager



Groups ofIsometries


What is aGroup?

GroupTerminology

Group Maps


Exercises


Examples of Groups of Isometries

Other Groups of Isometries

• The group E of all isometries of the plane

• The group O of all orthogonal matrices

ExerciseLet θ = 360/n where n is in the set N = {1, 2, 3, . . .} of naturalnumbers. Let r = R(θ).Show that set G = {rk | k = 0, 1, 2, 3, . . .} of nonnegativepowers of r forms a group with n elements.

WallpaperGroups

Lance Drager



Groups ofIsometries


What is aGroup?

GroupTerminology

Group Maps


Exercises


What is a Group?

DefinitionAbstractly, a group is a set G equipped with a binary operationG × G → G : (g , h) 7→ gh, that has the following properties.

Associative Law g(hk) = (gh)k for all g , h, k ∈ G

Identity There is an element e ∈ G so that ge = eg = gfor all g ∈ G . (This element turns out to beunique).

Inverses For every g ∈ G , there is an element h ∈ G sothat hg = gh = e. This element can be shown tobe unique and is denoted by h = g−1.

• Warning! The group operation need not be commutative,i.e., it may be that gh 6= hg .

WallpaperGroups

Lance Drager



Groups ofIsometries


What is aGroup?

GroupTerminology

Group Maps


Exercises


Group Terminology 1

• We usually just write a single letter like G for a group.The binary operation is not explicitly listed.

• If the group operation in G is commutative, we call G acommutative group or an abelian group.

• If G is commutative, we sometimes write the binaryoperation as +, in which case the identity element isdenoted by 0 and the inverse of g is denoted by −g .Which notation to use is a matter of choice. Example:The group of integers Z = {0,±1,±2,±3, . . .} withaddition.

WallpaperGroups

Lance Drager



Groups ofIsometries


What is aGroup?

GroupTerminology

Group Maps


Exercises


Group Terminology 2

DefinitionIf G is a group, a subset H ⊆ G is said to be a subgroup of G ifit is a group in its own right with the same operation as G , i.e.,

1 e ∈ H. (The identity is in H.)

2 h1, h2 ∈ H =⇒ h1h2 ∈ H (H is closed under takingproducts.)

3 h ∈ H =⇒ h−1 ∈ H (H is closed under taking inverses.)

• By a group of symmetries of the plane or a group ofisometries of the plane we mean a subgroup of E, thegroup of all isometries of the plane

WallpaperGroups

Lance Drager



Groups ofIsometries


What is aGroup?

GroupTerminology

Group Maps


Exercises


Group Terminology 3

• If G is a group, the subsets {e} ⊆ G and G ⊆ G arealways subgroups. If H ⊆ G is a subgroup, we say it is aproper subgroup if H $ G .

• If g ∈ G we can define the powers of g by• g0 = e.• If n ∈ N, then gn = gg . . . g︸︷︷︸

n factors

.

• If n is a negative integer, say n = −p, where p > 0, thengn = (g−1)p.

• If g ∈ G , the set 〈g〉 is a subgroup of G called the cyclicsubgroup of G generated by g .

WallpaperGroups

Lance Drager



Groups ofIsometries


What is aGroup?

GroupTerminology

Group Maps


Exercises


Group Maps

DefinitionLet G and H be groups. A transformation φ : G → H (it neednot be one-to-one or onto) is a Group Homomorphism orGroup Mapping if it satisfies

• φ(e) = e.

• φ(ab) = φ(a)φ(b) for all a, b ∈ G .

It follows that φ(g−1) = φ(g)−1.

DefinitionIf a ∈ G is fixed the mapping φa(g) = aga−1 is a group mapG → G called conjugation by a. Check:

ψa(e) = aea−1 = aa−1 = e

ψa(g)ψa(h) = (aga−1)(aha−1) = ageha−1 = a(gh)a−1 = ψa(gh)

WallpaperGroups

Lance Drager



Groups ofIsometries


What is aGroup?

GroupTerminology

Group Maps


Exercises


Group Maps and Subgroups 1

TheoremLet φ : G → H be a group map and let A ⊆ G be a subgroup.Then

φ(A) = {h ∈ H | h = φ(a) for some a ∈ A}

is a subgroup of A. In particular, φ(G ) is a subgroup of H.

Proof.We have to check the three properties of a subgroup.

(1.) e ∈ A, so e = φ(e) ∈ φ(A).

(2.) Suppose h1 and h2 are in φ(A). Then there are elementsa1, a2 ∈ A so that h1 = φ(a1) and h2 = φ(a2). But thenh1h2 = φ(a1)φ(a2) = φ(a1a2). Since a1a2 ∈ A, we haveh1h2 ∈ φ(A). Thus, φ(A) is closed under taking products.

WallpaperGroups

Lance Drager



Groups ofIsometries


What is aGroup?

GroupTerminology

Group Maps


Exercises



Proof Continued.

(3.) We have to show that φ(A) is closed under takinginverses. But, if h = φ(a) for some a ∈ A thenh−1 = φ(a)−1 = φ(a−1) ∈ φ(A), since a−1 ∈ A.

TheoremLet φ : G → H be a group map. Define ker(φ), the kernel of φby

ker(φ) = {g ∈ G | φ(g) = e}.

Then ker(φ) is a subgroup of G .

Proof.Exercise!

WallpaperGroups

Lance Drager



Groups ofIsometries


What is aGroup?

GroupTerminology

Group Maps


Exercises



DefinitionLet N be a subgroup of G . We say N is a normal subgroup ofG if

gng−1 ∈ N, for all n ∈ N and g ∈ G .

In other words, ψg (N) ⊆ N for all g ∈ G , where ψg is the mapgiven by conjugation by g . We can say that N is closed underconjugation by elements of G .

TheoremIf φ : G → H is a group map, then ker(φ) is a normal subgroupof G .

• If G is abelian, all subgroups of G are normal. There arelots of subgroups of nonabelian groups that are notnormal.

WallpaperGroups

Lance Drager



Groups ofIsometries


What is aGroup?

GroupTerminology

Group Maps


Exercises


Exercises on Groups 1

Exercises

• Let φ : G → H be a group map. Suppose that φ isone-to-one and onto, so there is an inverse mapφ−1 : H → G . Show that φ−1 is a group map. We saythat φ is a group isomorphism.

• We say that two groups G and H are isomorphic if there isa group isomorphism between them. This means thegroups can be put into one-to-one correspondence so thegroup operations match up.

• If a ∈ G , show that ψa is group isomorphism with

ψ−1a = ψa−1 .

• Let φ : G → H be a group map. Show ker(φ) is asubgroup of G and a normal subgroup of G .

WallpaperGroups

Lance Drager



Groups ofIsometries


What is aGroup?

GroupTerminology

Group Maps


Exercises


Exercises on Groups 2

Exercises Continued

• Let φ : G → H be a group map and let B ⊆ H be asubgroup of H. Show that

φ−1(B) = {g ∈ G | φ(g) ∈ B}

is a subgroup of G . Note ker(φ) = φ−1({e}).

• Find some subgroups of D3. Find some normal subgroupsof D3 and some nonnormal subgroups.

• Compute the conjugations ψr and ψs on D3. What kind ofelement do you get when you apply ψr to a rotation andto a reflection? What kind of element do you get whenyou apply ψs to rotations and reflections?

WallpaperGroups

Lance Drager



Groups ofIsometries


What is aGroup?

GroupTerminology

Group Maps


Exercises


Exercise on Groups 3

Harder Exercises for Later Thought

• What are the subgroups of Z? (Hint: every nonemptysubset of N has a smallest element.)

• Find, up to isomorphism, all groups that have 2, 3 and 4elements. Hint: Consider the possible multiplicationtables.

WallpaperGroups

Lance Drager



Groups ofIsometries


What is aGroup?

GroupTerminology

Group Maps


Exercises


Conjugation in E

• We divided the elements of E into 5 classes or types: theidentity, translations, rotations, reflections and glides.

• Every element T ∈ E can be written T = (A | a) where Ais an orthogonal matrix and a is a vector. A computationshows

(A | a)(B | b)(A | a)−1 = (ABA−1 | a + Ab − ABA−1a)

TheoremConjugation in E preserves type. In other words, if P,Q ∈ Ethen PQP−1 has the same type as Q.

• The proof is to just check all the cases. We omit thedetails, but some of them will come up later.

I insist on the details!

WallpaperGroups

Lance Drager

The Details ofConjugation inE

Conjugations in E, Part 1

• Conjugation by a translation (I | a)• Conjugating (I | b) gives (I | b) back. (Translations

commute.)• Conjugating a rotation translates the rotocenter by a.• Conjugating a Reflection [resp. glide] (S |αu + βv), where

Su = u and Sv = −v by (I | a) = (I | su + tv) gives(S |αu + βv + 2tv). This again a reflection [glide, sametranslation] with the mirror [glide] line shifted by tv in theperpendicular direction.

• Conjugation by a Rotation (R | a)• Conjugating (I | b) gives (I |Rb).• Conjugating (R ′ | p − R ′p) gives (R ′ |Rp − R ′(Rp)) i.e., it

changes the rotocenter from p to Rp, but does not theangle of rotation.

WallpaperGroups

Lance Drager



• Note in general that (A | a) = (I | a)(A | 0) and so

(A | a)(B | b)(A | b)−1 = (I | a)(A | 0)(B | b)(A | 0)−1(I | a)−1.

so it suffices to consider just conjugation by (A | 0).

• Conjugation by Rotation, continued.• Let S be a reflection matrix with Su = u and Sv = −v .

Set S ′ = RSR−1, u′ = Ru and v ′ = Rv . Then S ′u′ = u′

and S ′v ′ = −v ′. Thus, S ′ is a reflection matrix with themirror line rotated by R. If we have a reflection or glide(S |αu + βv) then conjugating by (R | 0) gives(RSR−1 |αRu + βRv) = (S ′ |αu′ + βv ′), which is of thesame type with the mirror or glide line rotated around theorigin.

WallpaperGroups

Lance Drager



• Conjugation by a reflection or guide.• If we conjugate (I | b) by (S | a) we get (I |Sb).• If R is a rotation matrix SRS = R−1. Hence conjugating

(R | b) by (S | 0) gives (R−1 |Sb) another rotation with theopposite angle. If p is the original rotocenter, the newrotocenter is Sp.

• Let T be a reflection matrix with Tx = x and Ty = −y ,let S be a reflection matrix, and let T ′ = STS , x ′ = Sxand y ′ = Sy . Then T ′x ′ = x ′ and T ′y ′ = −y ′. Thus T ′ isa reflection with the mirror transformed by S . If weconjugate (T |αx + βy) by (S | 0), we get(STS |αSx + βSy) = (T ′ |αx ′ + βy ′), which is of thesame type. So the mirror or glide line is transformed by S .

• Consider it an exercise to check the computations above.Good for the soul.

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UBS Businessplan. Strategic Planning and Financing Basis. Sample