112_Math 3121Abstract Algebra I

Math 3121
Abstract Algebra I
Lecture 8
Sections 9 and 10


Section 9
• Section 9: Orbits, Cycles, and the
Alternating Group
– Definition: Orbits of a permutation
– Definition: Cycle permutations
– Theorem: Every permutation of a finite set is a product of
disjoint cycles.
– Definition: Transposition
– Definition/Theorem: Parity of a permutation
– Definition: Alternating Group on n letters.


Orbits
• Look at what happens to elements as a
permutation is applied:
• Example:

1 2 3 4 5

α = 
 2 3 1 5 4


Orbits
Theorem: Let p be a permutation of a set S. The following relation is

an equivalence relation:
a~b

⇔

b = pn(a), for some n in ℤ

Proof:
1) reflexive: a = p0(a) ⇒ a~a
2) symmetric: a~b ⇒ b = pn(a), for some n in ℤ
⇒ a = p-n(b), with -n in ℤ
⇒ b~a
3) transitive: a~b and b~c
⇒ b = pn1(a) and c = pn2(b) , for some n1 and n2 in ℤ
⇒ c = pn2(pn1(a)) , for some n1 and n2 in ℤ
⇒ c = pn2+n1(a) , with n2 + n1 in ℤ
⇒ a~c


Definition: An orbit of a permutation p is an
equivalence class under the relation:
a~b

⇔

b = pn(a), for some n in ℤ


Example
• Find all orbits of

1 2 3 4 5

α = 
 2 3 1 5 4

• Method: Let S be the set that the permutation
works on. 0) Start with an empty list 1) If
possible, pick an element of the S not already
visited and apply permutation repeatedly to get
an orbit. 2) Repeat step 1 until all elements of S
have been visited.


Cycles
Definition: A permutation is a cycle if a most one
of its orbits is nontrivial (has more than one
element).
Notation: Cycle notation: list each orbit within
parentheses.
Example: Do this for
(1, 2, 3)(4, 5)

1 2 3 4 5

α = 
 2 3 1 5 4


Cycle Multiplication

• Examples: (without commas)
( 1 2 5 6 3) (1 3) =
(1 2 3) (1 2 3) =
(1 4)(1 3)(1 2) =


Cycle Decomposition
Theorem: Every permutation of a finite set is a
product of disjoint cycles.
Proof: Let σ be a permutation. Let B1, B2, … Br be
the orbits. Let μi be the cycle defined by
μi (x) = σ(x) if x in Bi and x otherwise
Then σ = μ1 μ2 … μr
Note: Disjoint cycles commute.


Examples
• Decompose S3 and make a multiplication table.


1
ρ 0 = 
1
1
ρ1 = 
2
1
ρ 2 = 
3
1

µ1 = 
1

2 3

2 3
2 3

3 1
2 3

1 2
2 3

3 2

1
µ 2 = 
3
1
µ3 = 
2

2 3

2 1
2 3

1 3

S3 = Symmetric Group on
3 Letters
ρ0

ρ1

ρ2

μ1

μ2

μ3

ρ0

ρ0

ρ1

ρ2

μ1

μ2

μ3

ρ1

ρ1

ρ2

ρ0

μ3

μ1

μ2

ρ2

ρ2

ρ0

ρ1

μ2

μ3

μ1

μ1

μ1

μ2

μ3

ρ0

ρ1

ρ2

μ2

μ2

μ3

μ1

ρ2

ρ0

ρ1

μ3

μ3

μ1

μ2

ρ1

ρ2

ρ0


Transpositions
Definition: A cycle of length 2 is called a
transposition:
Lemma: Every cycle is a product of
transpositions
Proof: Let (a1, a2, …, an) be a cycle, then
(a1, an) (a1, an-1) … (a1 a2) = (a1, a2, …, an)

Theorem: Every permutation can be written as
a product of transpositions.
Proof: Use the lemma plus the previous
theorem.


Parity of a Permutation

Definition: The parity of a permutation is
said to be even if it can be expressed as
the product of an even number of
transpositions, and odd if it can be
expressed as a product of an odd number

of transpositions.
Theorem: The parity of a permutation is
even or odd, but not both.


Parity of a Permutation

Definition: The parity of a permutation is said to be even if it can be
expressed as the product of an even number of transpositions,
and odd if it can be expressed as a product of an odd number of
permutations.
Theorem: The parity of a permutation is even or odd, but not both.
Proof: We show thatFor any positive integer n, parity is a
homomorphism from Sn to the group ℤ2, where 0 represents even,
and 1 represents odd. (These are alternate names for the
equivalence classes 2ℤ and 2ℤ+1 that make up the group ℤ2.


Defining the Parity Map
There are several ways to define the parity map. They tend to use the
group {1, -1} with multiplicative notation instead of {0, 1} with
additive notation.
One way uses linear algebra: For the permutation π define a map
from Rn to Rn by switching coordinates as follows Lπ(x1, x2, …, xn) = (x
π(1), xπ(2), …, xπ(n)). Then Lπ is represented by a nxn matrix Mπ whose
rows are the corresponding permutation of the rows of the nxn
identity matrix. The map that takes the permutation π to Det(Mπ) is
a homomorphism from Sn to the multiplicative group {-1, 1}.
Another way uses the action of the permutation on the polynomial
P(x1, x2, …, xn ) = Product{(xi - xj )| i < j }. Each permutation

changes the sign of P or leaves it alone. This determines the parity:
change sign = odd parity, leave sign = even parity.
Another way is to work directly with the cycles as in Proof2 in the
book.


Alternating Group
• Definition: The alternating group on n letters
consists of the even permutations in the
symmetric group of n letters.


HW Section 9
• Don’t hand in:
Pages 94-95: 19, 39
• Hand in Tues, Oct 28:
Pages 94-95: 10, 24, 36


Section 10
• Section 10: Cosets and the Theorem of Lagrange
– Modular relations for a subgroup
– Definition: Coset
– Theorem of Lagrange: For finite groups, the order of
subgroup divides the order of the group.
– Theorem: For finite groups, the order of any element
divides the order of the group


Modulo a Subgroup

Definition: Let H be a subgroup of a group G. Define
relations: ~L and ~R by:
x ~L y ⇔ x-1 y in H
x ~R y ⇔ x y-1 in H

We will show that ~L and ~R are equivalence relations on G.
We call ~L left modulo H.
We call ~R right modulo H.
•

Note:
x ~L y

⇔ x-1 y = h, for some h in H

x ~R y

⇔ y = x h, for some h in H
⇔ x y-1 = h, for some h in H
⇔ x = h y, for some h in H


Equivalence Modulo a Subgroup
Theorem: Let H be a subgroup of a group G. The relations: ~L and ~R defined by:
x ~L y ⇔ x-1 y in H
x ~R y ⇔ x y-1 in H

are equivalence relations on G.
Proof: We show the three properties for equivalence relations:
1) Reflexive: x-1 x = e is in H. Thus x ~L x.

2) Symmetric: x ~L y ⇒x-1 y in H
⇒ (x-1 y) -1 in H
⇒ y-1 x in H
⇒ y ~L x
3) Transitive: x ~L y and y ~L z ⇒ x-1 y in H and y-1 z in H
⇒ (x-1 y )( y-1 z) in H
⇒ (x-1 z) in H
⇒ x ~L z
Similarly, for x ~R y .


Cosets
• The equivalence classes for these equivalence relations
are called left and right cosets modulo the subgroup.
Recall: x ~L y
⇔ x-1 y = h, for some h in H
⇔ y = x h, for some h in H

• Cosets are defined as follows
Definition: Let H be a subgroup of a group G.
The subset
a H = { a h | h in H }

is called the left coset of H containing a, and the subset
H a= { a h | h in H }

is called the right coset of H containing a.


Examples

• Cosets of nℤ are:
nℤ, nℤ+1, nℤ+2, …, nℤ + (n-1
Note: Cosets in nonabelian case: left and right don’t
always agree.
• In the book: H = { ρ0, μ1} in S3 has different left
and right cosets.


Counting Cosets
Theorem: For a given subgroup of a group, every
coset has exactly the same number of elements,
namely the order of the subset.
Proof: Let H be a subgroup of a group G. Recall the
definitions of the cosets: aH and Ha.
a H = { a h | h in H }
H a= { a h | h in H }

Define a map La from H to aH by the formula La(g) = a g. This is 11 and onto.
Define a map Ra from H to Ha by the formula Ra(g) = g a. This is
1-1 and onto.


Lagrange
Theorem (Lagrange): Let H be a subgroup of a
finite group G. Then the order of H divides the
order of G.
Proof: Let n = number of left cosets of H, and let m
= the number of elements in H. Then n m = the
number of elements of G. Here m is the order of
H, and n m is the order of G.

Orders of Cycles
• The order of an element in a finite group is the
order of the cyclic group it generates. Thus the
order of any element divides the order of the
group.