Permutation Groups
Part 1
Definition

A permutation of a set A is a function
from A to A that is both one to one and
onto.

Array notation
Let A = {1, 2, 3, 4}
 Here are two permutations of A:
1 2 3 4
1 2 3 4
  
   

2 3 1 4
2 1 4 3

(2)  3
(4)  4
 (4)  3
  4
 (2)   (3)

 (1)  2
Composition in Array Notation
1 2 3 4 1 2 3 4
  
 

2 1 4 3 2 3 1 4
1 2 3 4
 

1


Composition in Array Notation
1
  
2
1
 
1
2 3 4 1 2 3 4
 

1 4 3 2 3 1 4
2 3 4

4

Composition in Array Notation
1
  
2
1
 
1
2 3 4 1 2 3 4
 

1 4 3 2 3 1 4
2 3 4

4 2

Composition in Array Notation
1
  
2
1
 
1
2 3 4 1 2 3 4
 

1 4 3 2 3 1 4
2 3 4
4 2 3 

Composition in Array Notation
1
  
2
1
 
1
2 3 4 1 2 3 4
 

1 4 3 2 3 1 4
2 3 4
4 2 3 

Definition

A permutation group of a set A is a set
of permutations of A that forms a group
under function composition.
Example


The set of all permutations on {1,2,3} is
called the symmetric group on three
letters, denoted S3
There are 6 permutations possible:
1 2
3 


__ __ __
3  2  1  6
S3

The permutations of {1,2,3}:
1 2 3
1 2 3 2 1 2 3
  
   
   

1 2 3
2 3 1
3 1 2
1 2 3
1 2 3 2
1 2 3
  
  
    

1 3 2
2 1 3
3 2 1


Is S3 a group?




Composition of functions is always
associative.
Identity is .
Since permutations are one to one and
onto, there exist inverses (which are
also permutations.
Therefore, S3 is group.
Computations in S3
1 2 3 1 2 3
  
 

2 3 1 3 1 2
3
1 2 3

 
1 2 3

1




1
2
3
1
2
3
2
  
 
 1
1 3 2 1 3 2 

1 2 3 1 2 3 1
  
 
 
1 3 2 2 3 1 3

2 3
 
2 3
2 3 2
  
2 1
Simplified computations in S3








Double the exponent of when
switching with .
You can simplify any expression in S3!
Symmetric groups, Sn



Let A = {1, 2, … n}. The symmetric
group on n letters, denoted Sn, is the
group of all permutations of A under
composition.
Sn is a group for the same reasons that
S3 is group.
|Sn| = n!
Symmetries of a square, D4
1
R0  
1
1
R90  
2
1
R180  
3
1
R270  
4
2 3 4

2 3 4
2 3 4

3 4 1
2 3 4

4 1 2
2 3 4

1 2 3
1 2 3 4
H  

2 1 4 3
1
V  
4
1
D  
1
2 3 4 3

3 2 1
2 3 4

4 3 2
4
1 2 3 4
D 

3 2 1 4
2
1
D4 ≤ S4
Why do we care?

Every group turns out to be a
permutation group on some set!
(To be proved later).
Cycle Notation
1 2 3 4
  

2 3 1 4
1 2 3 4
  

2 1 4 3
  (1 2 3)
  (1 2)(3 4)



Disjoint cycles


Two permutations are disjoint if the sets
of elements moved by the permutations
are disjoint.
Every permutation can be represented
as a product of disjoint cycles.
Algorithm for disjoint cycles



Let permutation π be given.
Let a be the identity permutation,
represented by an empty list of cycles.
while there exists n with π(n) ≠ a(n):
start a new cycle with n
let b = n
while
Compostion in cycle notation


 = (1 2 3)(1 2)(3 4)
= (1 3 4)(2)
= (1 3 4)
 = (1 2)(3 4)(1 2 3)
= (1)(2 4 3)
= (2 4 3)
Compostion in cycle notation


 = (1 2 3)(1 2)(3 4)
= (1
1 1 2 3
 = (1 2)(3 4)(1 2 3)