Download Math 3121 Abstract 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

  
 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

  
 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.
1
2
3
4
5


Example: Do this for   

 2 3 1 5 4


(1, 2, 3)(4, 5)
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
 0  
1
1
1  
2
1
 2  
3
2 3

2 3
2 3

3 1
2 3

1 2
1
1  
1
1
 2  
3
1
3  
2
2 3

3 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 ~R y
⇔ x-1 y = h, for some h in H
⇔ 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.
HW Section 10
• Don’t hand in:
– Pages 101: 3, 6, 9, 15
• Hand in Tues, Nov 4
– Pages 101-102: 8, 10