112_Math 3121Abstract Algebra I

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

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

1 3

2

5 4

3 2

1

α

1) reflexive: a = p 0 (a) ⇒ a~a

2) symmetric: a~b ⇒ b = p n (a), for some n in ℤ

⇒ a = p -n (b), with -n in ℤ

⇒ b~a 3) transitive: a~b and b~c

⇒ b = p n1 (a) and c = p n2 (b) , for some n1 and n2 in ℤ

⇒ c = p n2 (p n1 (a)) , for some n1 and n2 in ℤ

⇒ c = p n2+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

• Find all orbits of

• 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

1 3

2

5 4

3 2

1

α

Definition: A permutation is a cycle if a most one

of its orbits is nontrivial (has more than one

3 2

1

α

Cycle Decomposition

Theorem: Every permutation of a finite set is a

product of disjoint cycles

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

• Decompose S3 and make a multiplication table

123

321

23

1

32

1

21

3

32

1

132

321

321

321

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

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

identity matrix The map that takes the permutation π to Det(M π ) is

a homomorphism from S n to the multiplicative group {-1, 1}.

Another way uses the action of the permutation on the polynomial P(x 1 , x 2 , …, x n ) = Product{(x i - x j )| 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.

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

x ~ L y ⇔ x -1 y in H

x ~ R y ⇔ x y -1 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:

⇒ (x -1 y )( y -1 z) in H

⇒ (x -1 z) in H

⇒ x ~ L z

Similarly, for x ~ R y

• Cosets are defined as follows

Definition: Let H be a subgroup of a group G

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

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