Download Abstract Algebra

SECTION 8 Groups of Permutations
Definition
A permutation of a set A is a function ϕ: AA that is both one to one
and onto.
If and  are both permutations of a set A, then the composite function


 A 
 A gives a one-to-one and onto
   defined by A 
mapping of A into A.
We can show that function composition  is a binary operation, and call
this function composition  permutation multiplication. We will denote
  by  .
Remember that the action of   on A must be read in right-to-left order:
first apply  and then .
Notations
Example:
Suppose A = {1, 2, 3, 4, 5} and that  is the permutation given by
1 4, 2 2, 3 5, 4 3, 5 1. We can write  as following:
1 2 3 4 5 


4
2
5
3
1


 , then =
Let   

 3 5 4 2 1
12345
1 2 3 4 5 1 2 3 4 5  1 2 3 4 5 




4
2
5
3
1
3
5
4
2
1
5
1
3
2
4


 

Permutation Groups
Theorem
Le A be a nonempty set, and let SA be the collection of all permutations
of A. Then SA is a group under permutation multiplication.
Proof: exercise.
Symmetric Groups
Note: here we will focus on the case where A is finite. it’s also customary
to take A to be set of the form {1, 2, 3, …, , n} for some positive
integer n.
Definition:
Let A be the finite set {1, 2,   , n}. The group of all permutations of A is
the symmetric group on n letters , and is denoted by Sn.
Note that Sn has n! elements, where n!=n(n-1)(n-2)   (3)(2)(1).
Two important examples
Example: S3
Let set A be {1, 2, 3}. Then S3 is a group with 3!=6 elements. Let
1 2 3
1
,


 1 1
1
2
3



 1 2 3
1
1  
 ,  2  3
2
3
1



1 2 3 
1
2  
,


 3 2
3
1
2



0  
2 3
,

3 2
2 3
,
2 1
2 3
.
1 3
Then the multiplication table for S3 is shown in the next slide.
0
1
2
1
2
0
0
1
2
1
2
3
3
1
1  2
0
3
1
2
2
2
0
1
2
3
1
1
1
2
3
0
1
2
2 2
3
1
2
0
1
3 3
1
2
1
2
0
S3 and D3
Note that this group is not abelian ( 1u1  u11 )
There is a natural correspondence between the elements of S3 and the
ways in which two copies of an equilateral triangle with vertices 1, 2,
and 3 can be placed, one covering the other with vertices on to of
vertices.
For this reason, S3 is also the group D3 of symmetries of an equilateral
triangle. Naively, we used i for rotations and i for mirror images in
bisectors of angles.
3
1
2
Cayley’s Theorem
Definition
Let f: A B be a function and let H be a subset of A. The image of H
under f is { f (h) | h  H } and is denoted by f [H].
Lemma
Let G and G’ be groups and let : G G’ be a one-to-one function such
that  (x y) =  (x)  (y) for all x, y  G. Then  [G] is a subgroup of
G’ and  provides an isomorphism of G with  [G].
Then apply the above Lemma, we can show
Theorem (Cayley’s Theorem)
Every group is isomorphic to a group of permutations.