Wallpaper
Groups
Lance Drager
Introduction
to Groups of
Isometries
Symmetries of
an Equilateral
Triangle
Groups of
Isometries
A Little Group
Theory
What is a
Group?
Group
Terminology
Group Maps
Group Maps and
Subgroups
Exercises
Conjugation in
the Group of
Plane Isometries
Geometric Transformations and Wallpaper
Groups
Groups of Isometries
Lance Drager
Department of Mathematics and Statistics
Texas Tech University
Lubbock, Texas
2010 Math Camp
Wallpaper
Groups
Outline
Lance Drager
Introduction
to Groups of
Isometries
Symmetries of
an Equilateral
Triangle
Groups of
Isometries
A Little Group
Theory
What is a
Group?
Group
Terminology
Group Maps
Group Maps and
Subgroups
Exercises
Conjugation in
the Group of
Plane Isometries
1 Introduction to Groups of Isometries
Symmetries of an Equilateral Triangle
Groups of Isometries
2 A Little Group Theory
What is a Group?
Group Terminology
Group Maps
Group Maps and Subgroups
Exercises
Conjugation in the Group of Plane Isometries
Wallpaper
Groups
Symmetries of a Triangle
Lance Drager
Introduction
to Groups of
Isometries
Symmetries of
an Equilateral
Triangle
Groups of
Isometries
• Problem. Find all the symmetries of an equilateral
triangle.
C
A Little Group
Theory
What is a
Group?
Group
Terminology
Group Maps
Group Maps and
Subgroups
Exercises
Conjugation in
the Group of
Plane Isometries
A
B
Wallpaper
Groups
All the Symmetries
Lance Drager
Introduction
to Groups of
Isometries
Symmetries of
an Equilateral
Triangle
Groups of
Isometries
• There are six, the identity, (call it e), rotation by 120◦ ,
call it r , r 2 and the reflections SA , SB and SC = s in the
mirror lines MA , MB and MC
C
A Little Group
Theory
What is a
Group?
Group
Terminology
Group Maps
Group Maps and
Subgroups
Exercises
Conjugation in
the Group of
Plane Isometries
MB
MA
r
A
MC
B
Wallpaper
Groups
Comparing Products 1
Lance Drager
Introduction
to Groups of
Isometries
• Here’s SB .
C
Symmetries of
an Equilateral
Triangle
Groups of
Isometries
A
SB
A Little Group
Theory
What is a
Group?
Group
Terminology
Group Maps
Group Maps and
Subgroups
Exercises
Conjugation in
the Group of
Plane Isometries
A
B
C
B
• Here’s rs.
C
C
A
r
s
A
B
B
A
C
B
Wallpaper
Groups
Comparing Products 2
Lance Drager
Introduction
to Groups of
Isometries
• Here’s SA .
C
Symmetries of
an Equilateral
Triangle
Groups of
Isometries
B
SA
A Little Group
Theory
What is a
Group?
Group
Terminology
Group Maps
Group Maps and
Subgroups
Exercises
Conjugation in
the Group of
Plane Isometries
A
B
A
C
• Here’s r 2 s.
C
C
B
r2
s
A
B
B
A
A
C
Wallpaper
Groups
Find the Product
Lance Drager
Introduction
to Groups of
Isometries
• What is sr ?
Symmetries of
an Equilateral
Triangle
Groups of
Isometries
C
B
A Little Group
Theory
What is a
Group?
Group
Terminology
Group Maps
Group Maps and
Subgroups
Exercises
Conjugation in
the Group of
Plane Isometries
B
r
A
B
s
C
A
A
C
• Answer: SA = r 2 s
C
B
SA
A
B
A
C
Wallpaper
Groups
Lance Drager
Multiplication Table
Introduction
to Groups of
Isometries
Symmetries of
an Equilateral
Triangle
Groups of
Isometries
A Little Group
Theory
What is a
Group?
Group
Terminology
Group Maps
Group Maps and
Subgroups
Exercises
Conjugation in
the Group of
Plane Isometries
• The set of symmetries is D3 = {e, r , r 2 , s, rs, r 2 s} and we
have the relations r 3 = e, s 2 = e and sr = r 2 s.
• Multiplication table: (Row label) × (Column label)
e
r
r2
s
rs r 2 s
2
e
e
r
r
s
rs r 2 s
r
r2
e
rs r 2 s s
r
2
2
r
r
e
r r 2s s
rs
2
s
s r s rs
e
r2
r
2
rs
rs
s r s r
e
r2
2
2
2
r s r s rs
s
r
r
e
Wallpaper
Groups
Lance Drager
Introduction
to Groups of
Isometries
Relations and Multiplication
Examples
Symmetries of
an Equilateral
Triangle
Groups of
Isometries
A Little Group
Theory
What is a
Group?
Group
Terminology
Group Maps
Group Maps and
Subgroups
Exercises
Conjugation in
the Group of
Plane Isometries
• The table is completely determined by the relations
r 3 = e, s 2 = e and sr = r 2 s.
• Examples from the Multiplication Table
• Example: (r 2 s)(rs) = r 2 srs = r 2 (sr )s = r 2 (r 2 s)s = r 4 s 2 =
r 4 e = rr 3 = re = r .
• Example: (rs)(r 2 s) = rsr 2 s = r (sr )rs = r (r 2 s)rs = r 3 srs =
srs = (sr )s = (r 2 s)s = r 2 s 2 = r 2 .
Wallpaper
Groups
Groups of Isometries
Lance Drager
Introduction
to Groups of
Isometries
Symmetries of
an Equilateral
Triangle
Groups of
Isometries
A Little Group
Theory
What is a
Group?
Group
Terminology
Group Maps
Group Maps and
Subgroups
Exercises
Conjugation in
the Group of
Plane Isometries
• D3 is a group of isometries. In math, this means three
things
1
2
3
D3 contains the identity.
D3 is closed under taking products
D3 is closed under taking inverses.
• Dn is the group of symmetries of a regular n-gon. It is
generated by a rotation r and a reflection s, with the
relations r n = e and sr = r n−1 s. It has 2n elements.
Exercise
Verify that the group of symmetries of a square is D4 . If you
have 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.)
Wallpaper
Groups
Examples of Groups of Isometries
Lance Drager
Introduction
to Groups of
Isometries
Symmetries of
an Equilateral
Triangle
Groups of
Isometries
A Little Group
Theory
What is a
Group?
Group
Terminology
Group Maps
Group Maps and
Subgroups
Exercises
Conjugation in
the Group of
Plane Isometries
Other Groups of Isometries
• The group E of all isometries of the plane
• The group O of all orthogonal matrices
Exercise
Let θ = 360/n where n is in the set N = {1, 2, 3, . . .} of natural
numbers. Let r = R(θ).
Show that set G = {r k | k = 0, 1, 2, 3, . . .} of nonnegative
powers of r forms a group with n elements.
Wallpaper
Groups
What is a Group?
Lance Drager
Introduction
to Groups of
Isometries
Symmetries of
an Equilateral
Triangle
Groups of
Isometries
A Little Group
Theory
What is a
Group?
Group
Terminology
Group Maps
Group Maps and
Subgroups
Exercises
Conjugation in
the Group of
Plane Isometries
Definition
Abstractly, a group is a set G equipped with a binary operation
G × 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 = g
for all g ∈ G . (This element turns out to be
unique).
Inverses For every g ∈ G , there is an element h ∈ G so
that hg = gh = e. This element can be shown to
be 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 .
Wallpaper
Groups
Lance Drager
Group Terminology 1
Introduction
to Groups of
Isometries
Symmetries of
an Equilateral
Triangle
Groups of
Isometries
A Little Group
Theory
What is a
Group?
Group
Terminology
Group Maps
Group Maps and
Subgroups
Exercises
Conjugation in
the Group of
Plane Isometries
• 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 a
commutative group or an abelian group.
• If G is commutative, we sometimes write the binary
operation as +, in which case the identity element is
denoted 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, . . .} with
addition.
Wallpaper
Groups
Group Terminology 2
Lance Drager
Introduction
to Groups of
Isometries
Symmetries of
an Equilateral
Triangle
Groups of
Isometries
A Little Group
Theory
What is a
Group?
Group
Terminology
Group Maps
Group Maps and
Subgroups
Exercises
Conjugation in
the Group of
Plane Isometries
Definition
If G is a group, a subset H ⊆ G is said to be a subgroup of G if
it 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 =⇒ h1 h2 ∈ H (H is closed under taking
products.)
3
h ∈ H =⇒ h−1 ∈ H (H is closed under taking inverses.)
• By a group of symmetries of the plane or a group of
isometries of the plane we mean a subgroup of E, the
group of all isometries of the plane
Wallpaper
Groups
Group Terminology 3
Lance Drager
Introduction
to Groups of
Isometries
Symmetries of
an Equilateral
Triangle
Groups of
Isometries
A Little Group
Theory
What is a
Group?
Group
Terminology
Group Maps
Group Maps and
Subgroups
Exercises
Conjugation in
the Group of
Plane Isometries
• If G is a group, the subsets {e} ⊆ G and G ⊆ G are
always subgroups. If H ⊆ G is a subgroup, we say it is a
proper subgroup if H $ G .
• If g ∈ G we can define the powers of g by
• g 0 = e.
• If n ∈ N, then g n = gg . . . g .
| {z }
n factors
• If n is a negative integer, say n = −p, where p > 0, then
g n = (g −1 )p .
• If g ∈ G , the set hg i is a subgroup of G called the cyclic
subgroup of G generated by g .
Wallpaper
Groups
Group Maps
Lance Drager
Introduction
to Groups of
Isometries
Symmetries of
an Equilateral
Triangle
Groups of
Isometries
A Little Group
Theory
What is a
Group?
Group
Terminology
Group Maps
Group Maps and
Subgroups
Exercises
Conjugation in
the Group of
Plane Isometries
Definition
Let G and H be groups. A transformation φ : G → H (it need
not be one-to-one or onto) is a Group Homomorphism or
Group 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 .
Definition
If a ∈ G is fixed the mapping φa (g ) = aga−1 is a group map
G → 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)
Wallpaper
Groups
Group Maps and Subgroups 1
Lance Drager
Introduction
to Groups of
Isometries
Symmetries of
an Equilateral
Triangle
Groups of
Isometries
A Little Group
Theory
What is a
Group?
Group
Terminology
Group Maps
Group Maps and
Subgroups
Exercises
Conjugation in
the Group of
Plane Isometries
Theorem
Let 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 elements
a1 , a2 ∈ A so that h1 = φ(a1 ) and h2 = φ(a2 ). But then
h1 h2 = φ(a1 )φ(a2 ) = φ(a1 a2 ). Since a1 a2 ∈ A, we have
h1 h2 ∈ φ(A). Thus, φ(A) is closed under taking products.
Wallpaper
Groups
Group Maps and Subgroups 2
Lance Drager
Introduction
to Groups of
Isometries
Symmetries of
an Equilateral
Triangle
Groups of
Isometries
A Little Group
Theory
What is a
Group?
Group
Terminology
Group Maps
Group Maps and
Subgroups
Exercises
Conjugation in
the Group of
Plane Isometries
Proof Continued.
(3.) We have to show that φ(A) is closed under taking
inverses. But, if h = φ(a) for some a ∈ A then
h−1 = φ(a)−1 = φ(a−1 ) ∈ φ(A), since a−1 ∈ A.
Theorem
Let φ : G → H be a group map. Define ker(φ), the kernel of φ
by
ker(φ) = {g ∈ G | φ(g ) = e}.
Then ker(φ) is a subgroup of G .
Proof.
Exercise!
Wallpaper
Groups
Group Maps and Subgroups 3
Lance Drager
Introduction
to Groups of
Isometries
Symmetries of
an Equilateral
Triangle
Groups of
Isometries
A Little Group
Theory
What is a
Group?
Group
Terminology
Group Maps
Group Maps and
Subgroups
Exercises
Conjugation in
the Group of
Plane Isometries
Definition
Let N be a subgroup of G . We say N is a normal subgroup of
G 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 map
given by conjugation by g . We can say that N is closed under
conjugation by elements of G .
Theorem
If φ : G → H is a group map, then ker(φ) is a normal subgroup
of G .
• If G is abelian, all subgroups of G are normal. There are
lots of subgroups of nonabelian groups that are not
normal.
Wallpaper
Groups
Exercises on Groups 1
Lance Drager
Introduction
to Groups of
Isometries
Symmetries of
an Equilateral
Triangle
Groups of
Isometries
A Little Group
Theory
What is a
Group?
Group
Terminology
Group Maps
Group Maps and
Subgroups
Exercises
Conjugation in
the Group of
Plane Isometries
Exercises
• Let φ : G → H be a group map. Suppose that φ is
one-to-one and onto, so there is an inverse map
φ−1 : H → G . Show that φ−1 is a group map. We say
that φ is a group isomorphism.
• We say that two groups G and H are isomorphic if there is
a group isomorphism between them. This means the
groups can be put into one-to-one correspondence so the
group operations match up.
• If a ∈ G , show that ψa is group isomorphism with
−1
ψa = ψa−1 .
• Let φ : G → H be a group map. Show ker(φ) is a
subgroup of G and a normal subgroup of G .
Wallpaper
Groups
Exercises on Groups 2
Lance Drager
Introduction
to Groups of
Isometries
Symmetries of
an Equilateral
Triangle
Groups of
Isometries
A Little Group
Theory
What is a
Group?
Group
Terminology
Group Maps
Group Maps and
Subgroups
Exercises
Conjugation in
the Group of
Plane Isometries
Exercises Continued
• Let φ : G → H be a group map and let B ⊆ H be a
subgroup 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 subgroups
of D3 and some nonnormal subgroups.
• Compute the conjugations ψr and ψs on D3 . What kind of
element do you get when you apply ψr to a rotation and
to a reflection? What kind of element do you get when
you apply ψs to rotations and reflections?
Wallpaper
Groups
Lance Drager
Exercise on Groups 3
Introduction
to Groups of
Isometries
Symmetries of
an Equilateral
Triangle
Groups of
Isometries
A Little Group
Theory
What is a
Group?
Group
Terminology
Group Maps
Group Maps and
Subgroups
Exercises
Conjugation in
the Group of
Plane Isometries
Harder Exercises for Later Thought
• What are the subgroups of Z? (Hint: every nonempty
subset of N has a smallest element.)
• Find, up to isomorphism, all groups that have 2, 3 and 4
elements. Hint: Consider the possible multiplication
tables.
Wallpaper
Groups
Conjugation in E
Lance Drager
Introduction
to Groups of
Isometries
Symmetries of
an Equilateral
Triangle
Groups of
Isometries
A Little Group
Theory
What is a
Group?
Group
Terminology
Group Maps
Group Maps and
Subgroups
Exercises
Conjugation in
the Group of
Plane Isometries
• We divided the elements of E into 5 classes or types: the
identity, translations, rotations, reflections and glides.
• Every element T ∈ E can be written T = (A | a) where A
is an orthogonal matrix and a is a vector. A computation
shows
(A | a)(B | b)(A | a)−1 = (ABA−1 | a + Ab − ABA−1 a)
Theorem
Conjugation in E preserves type. In other words, if P, Q ∈ E
then PQP −1 has the same type as Q.
• The proof is to just check all the cases. We omit the
details, but some of them will come up later.
I insist on the details!
Wallpaper
Groups
Lance Drager
The Details of
Conjugation in
E
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, same
translation] with the mirror [glide] line shifted by tv in the
perpendicular direction.
• Conjugation by a Rotation (R | a)
• Conjugating (I | b) gives (I | Rb).
• Conjugating (R 0 | p − R 0 p) gives (R 0 | Rp − R 0 (Rp)) i.e., it
changes the rotocenter from p to Rp, but does not the
angle of rotation.
Wallpaper
Groups
Lance Drager
The Details of
Conjugation in
E
Conjugations in E, Part 2
• 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 0 = RSR −1 , u 0 = Ru and v 0 = Rv . Then S 0 u 0 = u 0
and S 0 v 0 = −v 0 . Thus, S 0 is a reflection matrix with the
mirror 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 0 | αu 0 + βv 0 ), which is of the
same type with the mirror or glide line rotated around the
origin.
Wallpaper
Groups
Lance Drager
The Details of
Conjugation in
E
Conjugations in E, Part 3
• 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 the
opposite angle. If p is the original rotocenter, the new
rotocenter is Sp.
• Let T be a reflection matrix with Tx = x and Ty = −y ,
let S be a reflection matrix, and let T 0 = STS, x 0 = Sx
and y 0 = Sy . Then T 0 x 0 = x 0 and T 0 y 0 = −y 0 . Thus T 0 is
a reflection with the mirror transformed by S. If we
conjugate (T | αx + βy ) by (S | 0), we get
(STS | αSx + βSy ) = (T 0 | αx 0 + βy 0 ), which is of the
same type. So the mirror or glide line is transformed by S.
• Consider it an exercise to check the computations above.
Good for the soul.