MATH 895-4 Fall 2010
Course Schedule
f a cu lty of science
d epa r tm ent of m athema tic s
Week
Date
Sections
from FS2009
1
Sept 7
I.1, I.2, I.3
2
14
I.4, I.5, I.6
3
21
II.1, II.2, II.3
4
28
II.4, II.5, II.6
5
Oct 5
III.1, III.2
6
12
IV.1, IV.2
L ECTURE 22
Part/ References
Topic/Sections
Combinatorial
Structures
FS: Part A.1, A.2
Comtet74
Handout #1
(self study)
Symbolic methods
Combinatorial
parameters
FS A.III
(self-study)
Combinatorial
Parameters
O RBIT-S TABILIZER T HEOREM
Notes/Speaker
Lecture 22:
Symmetry & Counting I:
The Orbit-Stabilizer Theorem
Contents
7
19
IV.3, IV.4
Unlabelled structures
Labelled structures I
Labelled structures II
Multivariable GFs
Complex Analysis
Analytic Methods
FS: Part B: IV, V, VI
Singularity Analysis
Appendix B4
22.1 Orbits
& Stablizers
. . . . . . . . . . . . . .
IV.5 V.1
Stanley 99: Ch. 6
9
Nov 2
Asymptotic methods
Handout #1
22.2 Permutations (self-study)
Acting on Sets: Application of
9
VI.1
8
10
11
12
13
Asst #1 Due
26
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
the Orbit-Stabilizer Theorem . . . . . . . . . . . . . .
5
12
22.2.1
. . . Mariolys
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.3/ CRotation Group of a Tetrahedron
Introduction to Prob.
5
18
IX.1 Rotation Group of a Cube
Limit Laws
22.2.2
. .and
. Comb
. . . . Marni
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
20
IX.2
Discrete Limit Laws
Sophie
22.2.3
Rotation
andGroup
Limit Lawsof an Octahedron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Asst #2 Due
Sophie
Random Structures
Combinatorial
Dodecahedron
instances of discrete
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
25
IX.4 Rotation Group of an Icosahedron
Continuous Limit Laws
22.2.5
. . Marni
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
an
IX.5 Rotation Group of an
22.2.6
1422.3
DecExercises
10
Mariolys
7
22.2.4
30
IX.3
FS: Part C
Rotation
Group of
(rotating
presentations)
23
Quasi-Powers and
Sophie
Soccer
Basket
Ball,
Gaussian Ball,
limit laws
Volley Ball, and Tennis Ball . . . . . . . . .
9
. . . . . .Presentations
. . . . . . . . . . . . . . . Asst
. .#3. Due
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
In this lecture we discuss how to use group theory to count like a professional: we look at an application of
cosets to determine the size of a permutation group. In particular, we discover a straightforward way to count
the number of symmetries of various geometric objects.
22.1
Orbits & Stablizers
Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
Version of: 11-Dec-09
In this section we will take a look at how permutation groups act on various structures.
It will be helpful to extend the definition of a permutation from finite sets of numbers Zn , to arbitrary sets.
Let X be a nonempty set. A permutation α of X is a bijection α : X → X. The set of all permutations of X is
called the symmetric group of X and is denoted by SX :
SX = {α | α : X → X is a bijection}.
If X = Zn = {1, 2, . . . , n} then we simply denoted SZn by Sn .
Definition 22.1 (Stabilizer of a Point) Let G be a subgroup of SX . For each i ∈ X, let
stabG (i) = {α ∈ G | α(i) = i}.
We call stabG (i) the stabilizer of i in G.
We can check that stabG (i) is a subgroup of G. Since ε fixes every element in X it is definitely in stabG (i). Let
α, β ∈ G, then α(i) = i and β(i) = i. It then follows that α−1 (i) = i and (αβ)(i) = β(α(i)) = β(i) = i, hence
α−1 , αβ ∈ stabG (i). Therefore stabG (i) < G.
Jamie Mulholland, Spring 2011
Math 302
22-1
MATH 895-4 Fall 2010
Course Schedule
f a cu lty of science
d epa r tm ent of m athema tic s
Week
Date
1
Sept 7
Sections
L ECTURE 22
Part/ References
Topic/Sections
O RBIT-S TABILIZER T HEOREM
Notes/Speaker
from FS2009
Definition 22.2
(Orbit of a Point) Let G be a subgroup of SX . For each i ∈ X, let
I.1, I.2, I.3
Combinatorial
Structures
2
14
I.4, I.5, I.6
FS: Part A.1, A.2
We call orbG (i) the orbit
of i under
Comtet74
3
21
II.1, II.2, II.3
Handout #1
(self study)
4
28
II.4, II.5, II.6
5
Oct 5
III.1, III.2
7
19
IV.3, IV.4
8
26
9
Nov 2
Combinatorial
Symbolic methods
orbG (i) = {α(i) | α ∈ G}.
Unlabelled structures
G.
Labelled structures I
Labelled structures II
Combinatorial
Asst #1 Due
Example 22.1 If G =parameters
S4 , then stabS4Parameters
(3) is the set of all permutation in S4 which fixes 3. There are 4! = 24
FS A.III
permutations
in
S
but
only
the
ones
that
don’tGFs
have 3 in their disjoint cycle form fix 3. Therefore,
6
12
IV.1, IV.2 4
Multivariable
(self-study)
IV.5 V.1
Notice we used the
10
9
VI.1
Complex
Analysis
stabS4 (3)
= {ε,
(1, 2), (1, 4), (2, 4), (1, 2, 4), (1, 4, 2)}
Analytic Methods
FS: Part B: IV, V, VI
Singularity
Analysis
= S{1,2,4} .
Appendix B4
Stanley 99: Ch. 6
Asst #2 Due
Asymptotic methods
Handout #1
notation
S
to denote
the set of all permutations of the set
(self-study) {1,2,4}
Sophie
12
A.3/ C
Example
22.2
Let
11
18
IX.1
20
IX.2
23
be12a group
25
13
30
14
Dec 10
Introduction to Prob.
Mariolys
Limit Laws and Comb
Marni
Discrete Limit Laws
Sophie
Combinatorial
instances of discrete
Mariolys
Continuous Limit Laws
Marni
G = h(1, 2, 3)(4, 5, 6)(7, 8)i
Random Structures
and2,
Limit
Laws
= {ε, (1,
3)(4,
5, 6)(7, 8), (1, 3, 2)(4, 6, 5), (7, 8), (1, 2, 3)(4, 5, 6), (1, 3, 2)(4, 6, 5)(7, 8)}.
FS: Part C
IX.3
(rotating
of permutation
on X
presentations)
IX.4
IX.5
{1, 2, 4}.
= {1, 2, 3, 4, 5, 6, 7, 8}. Then
orbG (1) = {1, 2, 3}
orbG (2) = {2, 3, 1}
Presentations
orbG (3) = {3,
1, 2}
orbG (4) = {4, 5, 6}
orbG (5) = {5, 6, 4}
orbG (6) = {6, 4, 5}
orbG (7) = {7, 8}
orbG (8) = {8, 7}
Quasi-Powers
and
stab
{ε, (7, 8)}
G (1) =
Sophie
Gaussian limit laws
stabG (2) = {ε, (7, 8)}
Asst
#3 (7,
Due 8)}
stabG (3) =
{ε,
stabG (4) = {ε, (7, 8)}
stabG (5) = {ε, (7, 8)}
stabG (6) = {ε, (7, 8)}
stabG (7) = {ε, (1, 2, 3)(4, 5, 6), (1, 3, 2)(4, 6, 5)}
stabG (8) = {ε, (1, 2, 3)(4, 5, 6), (1, 3, 2)(4, 6, 5)}
In each case notice that stabG (i) is a subgroup of G. Also notice that orbits are either disjoint or equal. Moreover,
the distinct orbits:
{1, 2, 3}, {4, 5, 6}, {7, 8}
Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
Version of: 11-Dec-09
form a partition of X.
Let G be a group of permutations on X, and define a relation on X by:
x ∼G y
⇐⇒
y = α(x) for some α ∈ G.
(1)
Then ∼G is an equivalence relation (see Exercise 1), and the equivalence class of an element x ∈ X is its orbit:
[x] = orbG (x).
Since equivalence classes partition the set, this indicates that our observation in Example 22.2 were not coincidence. Orbits will always be the same or disjoint, and distinct orbit classes will partition X.
Example 22.3 Recall that D4 , the dihedral group of the square, is the group of all symmetries of the square
(see Figure 1a). The elements are the rotations R0 , R90 , R180 , R270 , and the reflections H, V, D, D0 . We can view
D4 as a group of permutations on the vertices of the square. Here we identify the vertices of the square with the
set X = {1, 2, 3, 4}. See Figure 1b. Since vertex 1 can be taken to any other vertex by a rotation then the orbit of
1 is all of X: orbD4 (1) = {1, 2, 3, 4}.
The stabilizer of 1 is:
stabD4 (1) = {R0 , D}.
Similarly, we have stabD4 (2) = stabD4 (3) = {R0 , D0 }.
Jamie Mulholland, Spring 2011
Math 302
22-2
MATH 895-4 Fall 2010
Course Schedule
f a cu lty of science
d epa r tm ent of m athema tic s
Week
Date
1
Sept 7
I.1, I.2, I.3
2
14
I.4, I.5, I.6
3
21
II.1, II.2, II.3
4
28
II.4, II.5, II.6
5
Oct 5
III.1, III.2
6
12
IV.1, IV.2
7
19
8
26
9
Nov 2
Sections
from FS2009
IV.3, IV.4
IV.5 V.1
Part/ References
L ECTURE 22
Topic/Sections
Combinatorial
Structures
FS: Part A.1, A.2
Comtet74
Handout #1
(self study)
Symbolic methods
Combinatorial
parameters
FS A.III
(self-study)
Combinatorial
Parameters
Notes/Speaker
Unlabelled structures
Labelled structures I
Labelled structures II
Asst #1 Due
Multivariable GFs
(a) Reflection elements in D4
Complex Analysis
Analytic Methods
FS: Part B: IV, V, VI
Singularity
Analysis
Figure
D4 acting
as
Appendix1:
B4 The group
Stanley 99: Ch. 6
Asymptotic methods
Handout #1
(self-study)
(b) Orbit of vertex 1
a permutation group on the set of vertices.
Asst #2 Due
9
VI.1
12
A.3/ C
Introduction to Prob.
Mariolys
18
IX.1
Limit Laws and Comb
Marni
20
IX.2
Discrete Limit Laws
Sophie
23
IX.3
Combinatorial
instances of discrete
Mariolys
25
IX.4
Continuous Limit Laws
Marni
13
30
IX.5
14
Dec 10
10
11
12
Random Structures
and Limit Laws
FS: Part C
(rotating
presentations)
Quasi-Powers and
(a) Orbit of point P under
Gaussianaclimit laws
tion of D4
Presentations
O RBIT-S TABILIZER T HEOREM
Sophie
Sophie
Asst #3 Due
(b) Orbit of point Q under action of D4
Figure 2: The group D4 acting as a permutation group on the set of points enclosed by the square.
Example 22.4 Building on the previous example, we may view D4 as a group of permutations of the points X
enclosed by the square. Figure 2a illustrates the orbit of the point P and Figure 2b illustrates the orbit of the
point Q under D4 . Notice stabD4 (P ) = {R0 , D}, and stabD4 (Q) = {R0 }.
We can also view D as a group of permutations on the set of 4 line segments h, v, d, d0 shown in Figure 3. Then
Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
orbD4 (h) = {h, v}
stabD4 (h)
Version of: 11-Dec-09
orbD4 (v) = {h, v}
orbD4 (d) = {d, d0 }
orbD4 (d0 ) = {d, d0 }
= {R0 , R180 , H, V }
stabD4 (v) = {R0 , R180 , H, V }
stabD4 (d) = {R0 , R180 , D, D0 }
stabD4 (d0 ) = {R0 , R180 , D, D0 }
Figure 3: Orbit classes of the group D4 acting as a permutation group on the set of line segments h, v, d, d0 .
Example 22.5 Let RC3 be the Rubik’s cube group, and let X be the set of all cubies of Rubik’s cube. X can be
partitioned into edge cubies E, corner cubies V , and centre cubies C. If x denotes the uf edge cubie, then since it
is possible to move it to the location of any other edge cubie, then orbRC3 (x) = E. Also, since centre cubies don’t
move under cube moves, the orbit of each centre cubie is just a set of size 1.
Jamie Mulholland, Spring 2011
Math 302
22-3
MATH 895-4 Fall 2010
Course Schedule
f a cu lty of science
d epa r tm ent of m athema tic s
Week
Date
Sections
L ECTURE 22
Part/ References
Topic/Sections
O RBIT-S TABILIZER T HEOREM
Notes/Speaker
from FS2009
Example 22.6
Again, let RC3 be the Rubik’s cube group, but now let X be the set of all facets of Rubik’s cube.
Recall
|X|7 =I.1,48.
The Rubik’s cube group
canmethods
be viewed as a group of permutations of the set X (we have made
1
Sept
I.2, I.3
Symbolic
Combinatorial
use of this fact frequently
already). Let x be the facet on the up layer of the uf cubie. In our numbering system
Structures
2
14
I.4, I.5, I.6
Unlabelled structures
FS: Part
we denoted this facet by
x =A.1,7.A.2Since an edge cubie can be moved to the location of any other edge cubie, and
Comtet74
3
21
II.1, II.2, II.3
Labelled structures I
with
either
orientation,
then#1the orbit of
x is every edge-facet. Therefore, |orbRC3 (7)| = 24. The next theorem will
Handout
(self study)
|RC3 |
4
28
II.4,
II.5,
II.6
Labelled structures II
tell us that |stabRC3 (7)| = 24 .
5
Oct 5
III.1, III.2
Combinatorial
parameters
Combinatorial
Parameters
Asst #1 Due
Looking back at the examples
we can observe an obvious relationship between the sizes of G, orbG (i), and
FS A.III
6
12 weIV.1,
IV.2
Multivariable
GFs
(self-study)
stab
(i):
always
get
|orb
(i)|
· |stab
G
G
G (i)| equal to the size of G. This is true in general and is stated in the
next
theorem.
7
19
IV.3, IV.4
Complex Analysis
Analytic Methods
FS: Part B: IV, V, VI
Appendix B4
IV.5
V.1
Theorem 22.1 (Orbit-Stabilizer
Stanley 99: Ch. 6
9
Nov 2
Handout #1
(self-study)
9
VI.1
8
10
26
12
Singularity Analysis
Theorem) Let G be a subgroup of SX . Then for any i in X,
A.3/ C
Asst #2 Due
Asymptotic methods
|G| = |orbG (i)|
· |stabG (i)|.
Sophie
Introduction to Prob.
Mariolys
Proof:18 Since
stabG (x) is a subgroup of
G, we know from
Lagrange’s Theorem that
IX.1
Limit Laws and Comb
Marni
11
20
IX.2
Random Structures
|G|/|stab
number
right cosets of stabG (x) in G.
Discrete
Limit Lawsof distinct
Sophie
G (x)| = the
and Limit Laws
FS: Part C
Mariolys the number of elements in orbG (x). To this end
So we23need IX.3
to show that
the number Combinatorial
of right
cosets equals
(rotating
instances
of discrete
12
presentations)
define
25
IX.4
Continuous Limit Laws Marni
ψ : {(stabG (x))α | α ∈ G} → orbG (x)
by13
14
30
Quasi-Powers and
Gaussian limit laws
IX.5
Dec 10
Sophie
ψ(stabG (x)
α)
Asst
#3 =
Dueα(x).
Presentations
Our goal is to show that ψ is a bijection.
(a) ψ is well defined. We have
stabG (x) α = stabG (x) β
=⇒
α = γβ for some γ ∈ stabG (x)
=⇒
α(x) = (γβ)(x) = β(γ(x))
=⇒
α(x) = β(x) since γ ∈ stabG (x).
Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
Version of: 11-Dec-09
(b) ψ is injective. Let α, β ∈ G, we have
ψ(stabG (x) α) = ψ(stabG (x) β)
=⇒
α(x) = β(x)
=⇒
β −1 (α(x)) = x
=⇒
(αβ −1 )(x) = x
=⇒
αβ −1 ∈ stabG (x)
=⇒
stabG (x) α = stabG (x) β.
(c) ψ is surjective. Let y ∈ orbG (x). Then for some α ∈ G we have y = α(x). Therefore,
ψ(stabG (x) α) = α(x) = y,
and so ψ is surjective.
Therefore ψ is a bijection, and so it follows that
|orbG (x)| = |{(stabG (x))α | α ∈ G}|
= the number of right cosets of stabG (x) in G
= |G|/|stabG (x)|,
Jamie Mulholland, Spring 2011
Math 302
22-4
MATH 895-4 Fall 2010
Course Schedule
f a cu lty of science
d epa r tm ent of m athema tic s
Week
Date
1
Sept 7
Sections
L ECTURE 22
Part/ References
from FS2009
which implies
Topic/Sections
Combinatorial
Structures
I.4, I.5, I.6
2 14
FS: Part A.1, A.2
Comtet74
3
21
II.1, II.2, II.3
We now consider a fewHandout
applications
#1
(self study)
4
28
II.4, II.5, II.6
5
22.2
6
Oct 5
Notes/Speaker
|G|methods
= |orbG (i)|
Symbolic
I.1, I.2, I.3
O RBIT-S TABILIZER T HEOREM
· |stabG (i)|.
Unlabelled structures
structures I
of Labelled
this theorem.
Labelled structures II
Combinatorial
Combinatorial
FS A.III
(self-study)
Multivariable GFs
III.1, III.2
Asst #1 Due
parameters
Parameters
Permutations
Acting on
Sets: Application
of the Orbit-Stabilizer Theorem
12
IV.1, IV.2
The
orbit-stabilizer
theorem (TheoremComplex
22.1)Analysis
is a counting theorem. It enables one to determine the number
7
19
IV.3, IV.4
Methods
of elements in a set. Analytic
We
will
now
see
how
this theorem will help us determine the number of rotational
FS: Part B: IV, V, VI
8
26
Singularity Analysis
Appendix B4
symmetries
of
some
familiar
3-dimensional
objects.
IV.5 V.1
9
Stanley 99: Ch. 6
Nov 2
Asst #2 Due
Asymptotic methods
Handout #1
For a object X we let G
X be the group of all rotational symmetries of X. That is, the set of all ways the object
(self-study)
9
VI.1
Sophie
can
10 be picked up, rotated, and placed back on a table in front of you, so that it looks as though it wasn’t moved.
12
A.3/
C
Introduction
to
Prob.
For each of the objects below we will determine |GX |. Mariolys
11
18
20
22.2.1
IX.1
IX.2
Random Structures
Limit Laws and Comb
Marni
Discrete Limit Laws
Sophie
Combinatorial
instances of discrete
Mariolys
Continuous Limit Laws
Marni
Quasi-Powers and
Gaussian limit laws
Sophie
Rotation Group
ofLaws
a Tetrahedron
and Limit
23
IX.3
12
Let
GT be the group
25
IX.4
13
30
IX.5
14
Dec 10
FS: Part C
(rotating
ofpresentations)
all rotational
symmetries of a regular tetrahedron.
Presentations
Asst #3 Due
(a)
(b)
Figure 4: regular tetrahedron.
Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
Version of: 11-Dec-09
Let VT be the set of 4 vertices of the tetrahedron, labeled as in Figure 4b. Then each rotation in GT induces a
permutation on VT . That is, each element of GT gives a permutation in SVT = S4 . Vertex 1 can be taken to any
other vertex by a rotation, so the orbit of vertex 1 is orbGT (1) = {1, 2, 3, 4}, and therefore |orbGT (1)| = 4. The
stabilizer of 1 consists satisfies |stabGT (1)| = 3, and the rotations in the stabilizer are: the identity, and two
rotations corresponding to the permutations (2, 3, 4) and (2, 4, 3). Therefore, by the orbit-stabilizer theorem:
|GT | = |orbGT (1)| · |stabGT (1)| = 4 · 3 = 12.
The 12 rotations of GT are shown in Figure 5. Each rotation is described by the permutation it induces on the
vertices. It is clear from this description that GT ≈ A4 .
22.2.2
Rotation Group of a Cube
Let GC be the group of all rotational symmetries of a cube.
We can view GC as a groups of permutations of the 8 corners, that is, as a subgroup of S8 . Observe that
orbGC (1) = {1, 2, 3, 4, 5, 6, 7, 8}
⇒
|orbGC (1)| = 8
and that
stabGC (1) = {ε, (2, 4, 5)(3, 8, 6), (2, 5, 4)(3, 6, 8)}
Jamie Mulholland, Spring 2011
Math 302
⇒
|stabGC (1)| = 3.
22-5
MATH 895-4 Fall 2010
Course Schedule
f a cu lty of science
d epa r tm ent of m athema tic s
Week
Date
1
Sept 7
I.1, I.2, I.3
2
14
I.4, I.5, I.6
3
21
II.1, II.2, II.3
4
28
II.4, II.5, II.6
5
Oct 5
III.1, III.2
6
12
IV.1, IV.2
7
19
8
26
9
Nov 2
Sections
from FS2009
(a) ε
IV.3, IV.4
IV.5 V.1
L ECTURE 22
Part/ References
Topic/Sections
Combinatorial
Structures
FS: Part A.1, A.2
Comtet74
Handout #1
(self study)
Symbolic methods
Combinatorial
parameters
FS A.III
(self-study)
Combinatorial
Parameters
Notes/Speaker
Unlabelled structures
Labelled structures I
Labelled structures II
Asst #1 Due
Multivariable GFs
(b) (1, 4)(2, 3)
Complex Analysis
Analytic Methods
FS: Part B: IV, V, VI
Appendix B4
Stanley 99: Ch. 6
Handout #1
(self-study)
Asymptotic methods
VI.1
A.3/ C
Introduction to Prob.
Mariolys
18
IX.1
Limit Laws and Comb
Marni
20
IX.2
Discrete Limit Laws
Sophie
23
IX.3
Combinatorial
instances of discrete
Mariolys
25
IX.4
Continuous Limit Laws
Marni
13
30
IX.5
Quasi-Powers and
Gaussian limit laws
Sophie
14
Dec 10
12
(e) (2, 3, 4)
(d) (1, 2)(3, 4)
(g) (1, 4, 3)
(h) (1, 3, 4)
(k) (1, 3, 2)
(l) (1, 2, 3)
Asst #2 Due
12
11
(c) (1, 3)(2, 4)
Singularity Analysis
9
10
O RBIT-S TABILIZER T HEOREM
Sophie
Random Structures
and Limit Laws
FS: Part C
(rotating
presentations)
(f) (2, 4, 3)
Presentations
(i) (1, 2, 4)
Asst #3 Due
(j) (1, 4, 2)
Figure 5: All 12 rotational symmetries of a regular tetrahedron
Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
Version of: 11-Dec-09
(a)
(b)
Figure 6: cube.
The elements of the stabilizer are the rotations about an axis through vertices 1 and 7.
Therefore, by the orbit stabilizer theorem:
|GC | = |orbGC (1)| · |stabGC (1)| = 8 · 3 = 24.
Recall the symmetric group S4 has 24 elements. Perhaps GC is S4 in disguise. To see if it is we should find 4
things in the cube that GC permutes. There are 4 diagonals as shown in Figure 7, and each rotation of the cube
permutes these diagonals. In fact, each rotation of the cube can be described precisely by how these diagonals
Jamie Mulholland, Spring 2011
Math 302
22-6
MATH 895-4 Fall 2010
Course Schedule
f a cu lty of science
d epa r tm ent of m athema tic s
Week
Date
Sections
1
Sept 7
I.1, I.2, I.3
2
14
I.4, I.5, I.6
3
21
II.1, II.2, II.3
4
28
II.4, II.5, II.6
5
Oct 5
III.1, III.2
6
12
IV.1, IV.2
7
19
IV.3, IV.4
8
26
9
Nov 2
L ECTURE 22
Part/ References
Topic/Sections
fromTherefore
FS2009
are permuted.
GC ≈ S4 .
IV.5 V.1
Combinatorial
Structures
FS: Part A.1, A.2
Comtet74
Handout #1
(self study)
Symbolic methods
Combinatorial
parameters
FS A.III
(self-study)
Combinatorial
Parameters
Analytic Methods
Complex Analysis
Notes/Speaker
Unlabelled structures
Labelled structures I
Labelled structures II
Asst #1 Due
Multivariable GFs
Figure
7:IV,Viewing
GC as a group of permutations on the diagonals 1, 2, 3, 4.
FS: Part B:
V, VI
Singularity Analysis
Appendix B4
Stanley 99: Ch. 6
Handout #1
(self-study)
Asymptotic methods
9
VI.1
22.2.3
Rotation
Group of an Octahedron
10
12
O RBIT-S TABILIZER T HEOREM
A.3/ C
Introduction to Prob.
Asst #2 Due
Sophie
Mariolys
Let GO18 be the
of a regular
octahedron.
IX.1 group of all rotational symmetries
Limit Laws and Comb
Marni
11
20
IX.2
23
IX.3
25
IX.4
13
30
IX.5
14
Dec 10
12
Random Structures
and Limit Laws
FS: Part C
(rotating
presentations)
Discrete Limit Laws
Sophie
Combinatorial
instances of discrete
Mariolys
Continuous Limit Laws
Marni
Quasi-Powers and
Gaussian limit laws
Sophie
Presentations
Asst #3 Due
(a)
(b)
Figure 8: regular octahedron.
We can view GO as a groups of permutations of the 6 vertices, that is as a subgroup of S6 . Observe that
Dr. Marni MISHNA, Department of Mathematics, SIMON
UNIVERSITY
orbGFRASER
(1) =
{1, 2, 3, 4, 5, 6}
O
Version of: 11-Dec-09
⇒
|orbGO (1)| = 6
and that
stabGO (1) = {ε, (2, 3, 4, 5), (2, 4)(3, 5), (2, 5, 4, 3)}
⇒
|stabGO (1)| = 4.
The elements of the stabilizer are the rotations about an axis through vertices 1 and 6.
Therefore, by the orbit stabilizer theorem:
|GC | = |orbGO (1)| · |stabGO (1)| = 6 · 4 = 24.
It is no coincidence that this is the same size as the group of symmetries of the cube. Figure 9 shows the
octahedron sitting inside the cube (join midpoints of every two squares by a line). This means that GC ≈ GO .
The cube and the octahedron are referred to as dual solids.
22.2.4
Rotation Group of an Dodecahedron
Let GD be the group of all rotational symmetries of a regular dodecahedron.
We can view GD as a groups of permutations of the 20 vertices, that is as a subgroup of S20 . Observe that
orbGD (1) = {1, 2, 3, . . . , 20}
Jamie Mulholland, Spring 2011
Math 302
⇒
|orbGD (1)| = 20
22-7
MATH 895-4 Fall 2010
Course Schedule
f a cu lty of science
d epa r tm ent of m athema tic s
Week
Date
1
Sept 7
I.1, I.2, I.3
2
14
I.4, I.5, I.6
3
21
II.1, II.2, II.3
4
28
II.4, II.5, II.6
5
Oct 5
III.1, III.2
6
12
IV.1, IV.2
7
19
IV.3, IV.4
8
26
9
Nov 2
10
11
12
13
14
Sections
from FS2009
IV.5 V.1
L ECTURE 22
Part/ References
Topic/Sections
Combinatorial
Structures
FS: Part A.1, A.2
Comtet74
Handout #1
(self study)
Symbolic methods
Combinatorial
parameters
FS A.III
(self-study)
Combinatorial
Parameters
Analytic Methods
FS: Part B:Figure
IV, V, VI
Appendix B4
Stanley 99: Ch. 6
Handout #1
(self-study)
Notes/Speaker
Unlabelled structures
Labelled structures I
Labelled structures II
Asst #1 Due
Multivariable GFs
Complex Analysis
9: The octahedron is dual to the cube, so GO ≈ GC .
Singularity Analysis
Asst #2 Due
Asymptotic methods
9
VI.1
12
A.3/ C
Introduction to Prob.
Mariolys
18
IX.1
Limit Laws and Comb
Marni
20
IX.2
Discrete Limit Laws
Sophie
23
IX.3
Combinatorial
instances of discrete
Mariolys
25
IX.4
Continuous Limit Laws
Marni
30
IX.5
Quasi-Powers and
Gaussian limit laws
Sophie
Random Structures
and Limit Laws
FS: Part C
(rotating
presentations)
Dec 10
O RBIT-S TABILIZER T HEOREM
Sophie
(a)
Presentations
Asst #3 Due
(b)
Figure 10: regular dodecahedron.
and that
|stabGD (1)| = 3.
The elements of the stabilizer are the rotations about an axis through vertices 1 and 18.
Therefore, by the orbit stabilizer theorem:
Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
Version of: 11-Dec-09
|GC | = |orbG (1)|
D
22.2.5
· |stabGD (1)| = 20 · 3 = 60.
Rotation Group of an Icosahedron
Let GI be the group of all rotational symmetries of a regular icosahedron.
(a)
(b)
Figure 11: regular icosahedron.
Jamie Mulholland, Spring 2011
Math 302
22-8
MATH 895-4 Fall 2010
Course Schedule
f a cu lty of science
d epa r tm ent of m athema tic s
Week
Date
Sections
1
Sept 7
I.1, I.2, I.3
2
14
I.4, I.5, I.6
L ECTURE 22
Part/ References
Topic/Sections
O RBIT-S TABILIZER T HEOREM
Notes/Speaker
We can viewfrom
GI FS2009
as a groups of permutations of the 12 vertices, that is as a subgroup of S20 . Observe that
and
3 that
21
4
28
II.1, II.2, II.3
II.4, II.5, II.6
Symbolic methods
Combinatorial
orbGI (1) = {1, 2, 3, . . . , 12}
Structures
Unlabelled structures
FS: Part A.1, A.2
Comtet74
Labelled structures I
Handout #1
|stabGI (1)|
(self study)
Labelled structures II
⇒
|orbGI (1)| = 12
= 5.
The elements of the stabilizer
are the Combinatorial
rotations about an axis through vertices 1 and 12.
Combinatorial
5
Oct 5
III.1, III.2
parameters
Parameters
Asst #1 Due
A.III
Therefore, by the orbitFSstabilizer
theorem:
6
12
7
19
8
26
IV.1, IV.2
Multivariable GFs
(self-study)
|GC |Complex
= |orb
Analysis
GI (1)| · |stabGI (1)| = 20 · 3 = 60.
IV.3, IV.4
Analytic Methods
FS: Part B: IV, V, VI
Singularity Analysis
Appendix B4
IV.5 V.1
It 9is noNovcoincidence
that
this
same
size as the group
of symmetries of a regular dodecahedron. Figure 12
Stanley
99:is
Ch.the
6
2
Asst #2 Due
Asymptotic methods
#1
shows the octahedron Handout
sitting
inside
the
cube
(join
midpoints
of every two squares by a line). This means that
(self-study)
9
VI.1
Sophie
GI10≈ GD .
12
A.3/ C
Introduction to Prob.
Mariolys
18
IX.1
Limit Laws and Comb
Marni
20
IX.2
Discrete Limit Laws
Sophie
23
IX.3
Combinatorial
instances of discrete
Mariolys
25
IX.4
Continuous Limit Laws
Marni
13
30
IX.5
Quasi-Powers and
Gaussian limit laws
Sophie
14
Dec 10
11
12
Random Structures
and Limit Laws
FS: Part C
(rotating
presentations)
Presentations
Asst #3 Due
Figure 12: The icosahedron is dual to the dodecahedron, so GI ≈ GD .
22.2.6
Rotation Group of an Soccer Ball, Basket Ball, Volley Ball, and Tennis Ball
The balls used in soccer, basketball, volleyball, and tennis have district patterns on their surface. We can use
the orbit-stabilizer theorem to determine the rotational groups of symmetries of these patterns.
Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
Version of: 11-Dec-09
(a) soccer ball
(b) basket ball
(c) volley ball
(d) tennis ball
Figure 13: Familiar sports balls.
For each ball, pick an object on the ball: either a point, or shape. Determine the size of the orbit and stabilizer
of the point/shape and verify the results in the Table 1.
It will help if you have a physical ball in your hands. For the soccer ball, there are 12 pentagons (the black
faces), and 20 hexagons. See Figure 14 for an unfolded view of the soccer ball.
In case you are interested, the rotational group of the soccer ball is A5 .
In nature, the helix is the structure that occurs most often. The second most commonly found structures are
polyhedrons made from pentagons and hexagons, such as the dodecahedron and the truncated icosahedron
Jamie Mulholland, Spring 2011
Math 302
22-9
MATH 895-4 Fall 2010
Course Schedule
f a cu lty of science
d epa r tm ent of m athema tic s
Week
Date
Sections
from FS2009
1
Sept 7
I.1, I.2, I.3
2
14
I.4, I.5, I.6
3
21
II.1, II.2, II.3
4
28
II.4, II.5, II.6
5
Oct 5
III.1, III.2
6
12
IV.1, IV.2
7
19
IV.3, IV.4
8
26
9
Nov 2
IV.5 V.1
L ECTURE 22
Part/ References
Topic/Sections
ball
Notes/Speaker
size of group of rotations
Symbolic methods
Combinatorial
soccer ball
Structures
Unlabelled structures
basket
ball
FS: Part A.1,
A.2
Comtet74
volley ballLabelled structures I
Handout #1
(self study)tennis ball
Labelled structures II
Combinatorial
parameters
Table 1: The
FS A.III
(self-study)
Combinatorial
60
4
12
4
Asst #1 Due
Parameters
size
of the rotational group for various playing balls.
Analytic Methods
FS: Part B: IV, V, VI
Appendix B4
Stanley 99: Ch. 6
Handout #1
(self-study)
Multivariable GFs
Complex Analysis
Singularity Analysis
Asymptotic methods
Asst #2 Due
9
VI.1
12
A.3/ C
Introduction to Prob.
Mariolys
18
IX.1
Limit Laws and Comb
Marni
20
IX.2
Discrete Limit Laws
Sophie
23
IX.3
Combinatorial
instances of discrete
Mariolys
25
IX.4
Continuous Limit Laws
Marni
13
30
IX.5
Quasi-Powers and
Gaussian limit laws
Sophie
14
Dec 10
10
11
12
O RBIT-S TABILIZER T HEOREM
Random Structures
and Limit Laws
FS: Part C
(rotating
presentations)
Sophie
Presentations
Asst #3 Due
Figure 14: A soccer ball unfolded.
(soccer ball). Although it is impossible to enclose a space with hexagons along, adding 12 pentagons will be
sufficient to enclose the space (like the soccer ball). Many viruses have this kind of structure (Figure 15). 1
Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
Version of: 11-Dec-09
(a) rhinovirus (common cold)
(b) Archaeal virus
Figure 15: Viruses.
1 John
Galloway, Nature’s Second-Favourite Structure. New Scientist 114 (March 1988); 36-39
Jamie Mulholland, Spring 2011
Math 302
22-10
MATH 895-4 Fall 2010
Course Schedule
f a cu lty of science
d epa r tm ent of m athema tic s
Week
Date
1
Sept 7
2
22.3
14
3
21
4
28
Sections
from FS2009
Part/ References
I.1, I.2, I.3
Combinatorial
Structures
I.4, I.5, I.6
FS: Part A.1, A.2
Comtet74
II.1, II.2, II.3
Handout #1
the relation
defined in
(self study)
II.4, II.5, II.6
Exercises
1. Prove
L ECTURE 22
Topic/Sections
O RBIT-S TABILIZER T HEOREM
Notes/Speaker
Symbolic methods
Unlabelled structures
Labelled structures I
(1) is an equivalence relation.
Labelled structures II
2. Let RC3 be the Rubik’s
cube group
and let H be the subgroup generated by the product α = U R.
Combinatorial
Combinatorial
5
Oct 5
III.1, III.2
6
12
IV.1, IV.2
7
parameters
FS A.III
(self-study)
Parameters
Multivariable GFs
Asst #1 Due
H = hU Ri.
19
IV.3, IV.4
Complex Analysis
Let
X be
the set Analytic
of all Methods
cubies of Rubik’s
cube.
8
26
9
Nov 2
FS: Part B: IV, V, VI
Appendix B4
Singularity Analysis
Handout #1
Asymptotic methods
V.1
(a) If IV.5
x denotes
the uf
corner
cubie, determine orbH (x).
Stanley
99:rCh.
6
Asst #2 Due
(b) If y denotes (self-study)
the uf edge cubie, determine orbH (y).
10
9
VI.1
Sophie
(c) How
many elements do stabIntroduction
stab (y) have?
H (x) and
A.3/ C
to Prob. H Mariolys
12
18
IX.1of considering the set ofLimit
Laws and of
Comb
3. Instead
vertices
the Marni
tetrahedron, consider how GT permutes the 6 edges of
the
tetrahedron.
By
picking
one
edge,
say
the
edge
Random Structures
20
IX.2
Discrete Limit Laws
Sophie 12, the edge between vertices 1 and 2, verify that
Limit
Laws
|orbGT (12)| · |staband
(12)|
=
12.
G
FS:TPart C
11
12
23
IX.3
Combinatorial
instances of discrete
(rotating
Mariolys
4. Consider how GTpresentations)
permutes the 3 triangular faces of the tetrahedron. That is, consider GT as a subgroup
25
Continuous Limit Laws Marni
of
S3 . IX.4
By picking one face, say the
face f1,2,3 containing vertices 1, 2 and 3, verify that |orbGT (f1,2,3 )| ·
Quasi-Powers
and
|stab
(f
)|
=
12.
GTIX.5 1,2,3
13
30
Sophie
Gaussian limit laws
5.
14
Instead
of considering Presentations
the set of vertices of the dodecadedron,
consider how GD permutes the 30 edges of
Dec 10
Asst #3 Due
the dodecahedron. That is, consider GD as a subgroup of S30 . By picking one edge, say the edge 12, the
edge between vertices 1 and 2, verify that |orbGD (12)| · |stabGD (12)| = 60.
6. Consider how GD permutes the 12 pentagonal faces of the dodecahedron. That is, consider GD as a
subgroup of S12 . By picking one face, say the face f containing vertices 1, 2, 3, 4, 5, verify that |orbGD (f )| ·
|stabGD (f )| = 60.
7. For each of the following objects, describe each element of the group of rotations as a single rotation.
(Similar to what was done for the tetrahedron in Figure 5.)
Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
cube
Version of:(a)
11-Dec-09
(b) octahedron
8. Let G be the group of rotations of a rectangular box of dimensions 1 × 2 × 3. Describe each element of G
as a rotation.
9. Let G be the group of rotations of a rectangular box of dimensions 1 × 2 × 2. Describe each element of G
as a rotation.
10. The group D4 acts as a group of permutations of the points enclosed by the square shown below. (The
axis of symmetry are drawn for reference purposes.) For each square, locate the points in the orbit of the
indicated point P under the action of D4 . In each case, determine the stabilizer of P .
(a)
Jamie Mulholland, Spring 2011
Math 302
(b)
(c)
22-11
MATH 895-4 Fall 2010
Course Schedule
f a cu lty of science
d epa r tm ent of m athema tic s
Week
Date
Sections
L ECTURE 22
Part/ References
Topic/Sections
O RBIT-S TABILIZER T HEOREM
Notes/Speaker
from
FS2009
11. A soccer
ball
has 20 faces that are regular hexagons and 12 faces that are regular pentagons (see Figures
13a
14).
the orbit stabilizer
theorem to explain why a soccer ball cannot have 60◦ rotational
1
Sept 7and
I.1, I.2,
I.3 Use
Symbolic methods
Combinatorial
symmetry about Structures
a line through the centres of two opposite hexagonal faces.
2
12.
3
4
14
I.4, I.5, I.6
FS: Part A.1, A.2
Comtet74
For
21 each
II.1, of
II.2, the
II.3 solids below,
Handout #1
also
shown
as “unfolded”.)
(self study)
28
II.4, II.5, II.6
5
Oct 5
III.1, III.2
6
12
IV.1, IV.2
7
19
IV.3, IV.4
8
26
9
Nov 2
IV.5 V.1
Unlabelled structures
determine
the number
of rotational symmetries. (In the figures each solid is
Labelled structures
I
Labelled structures II
Combinatorial
parameters
FS A.III
(self-study)
Combinatorial
Parameters
Analytic Methods
FS: Part B: IV, V, VI
Appendix B4
Stanley 99: Ch. 6
Handout #1
(self-study)
Complex Analysis
Asst #1 Due
Multivariable GFs
Singularity Analysis
Asymptotic methods
Asst #2 Due
9
VI.1
12
A.3/ C
Introduction to Prob.
18
IX.1
Limit Laws and Comb
Marni
20
IX.2
Discrete Limit Laws
Sophie
23
IX.3
Combinatorial
instances of discrete
Mariolys
25
IX.4
Continuous Limit Laws
Marni
13
30
IX.5
Quasi-Powers and
Gaussian limit laws
Sophie
14
Dec 10
10
11
12
Sophie
Mariolys
(a) cuboctahedron
Random Structures
and Limit Laws
FS: Part C
(rotating
presentations)
Presentations
Asst #3 Due
(b) (small) rhombicuboctahedron
Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
Version of: 11-Dec-09
(c) great rhombicuboctahedron or truncated cuboctahedron
Jamie Mulholland, Spring 2011
Math 302
22-12