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:

€

α =1 2 3 4

2 3 1 4

⎡

⎣ ⎢

⎤

⎦ ⎥

€

β =1 2 3 4

2 1 4 3

⎡

⎣ ⎢

⎤

⎦ ⎥

€

α(2) = 3

€

α(4) = 4

€

β(4) = 3

€

β(1) = 2

€

βα(2) = β (3) = 4

Composition in Array Notation

€

βα =1 2 3 4

2 1 4 3

⎡

⎣ ⎢

⎤

⎦ ⎥

1 2 3 4

2 3 1 4

⎡

⎣ ⎢

⎤

⎦ ⎥

=1 2 3 4 ⎡

⎣ ⎢

⎤

⎦ ⎥1

Composition in Array Notation

€

βα =1 2 3 4

2 1 4 3

⎡

⎣ ⎢

⎤

⎦ ⎥

1 2 3 4

2 3 1 4

⎡

⎣ ⎢

⎤

⎦ ⎥

=1 2 3 4 ⎡

⎣ ⎢

⎤

⎦ ⎥1 4

Composition in Array Notation

€

βα =1 2 3 4

2 1 4 3

⎡

⎣ ⎢

⎤

⎦ ⎥

1 2 3 4

2 3 1 4

⎡

⎣ ⎢

⎤

⎦ ⎥

=1 2 3 4 ⎡

⎣ ⎢

⎤

⎦ ⎥1 4 2

Composition in Array Notation

€

βα =1 2 3 4

2 1 4 3

⎡

⎣ ⎢

⎤

⎦ ⎥

1 2 3 4

2 3 1 4

⎡

⎣ ⎢

⎤

⎦ ⎥

=1 2 3 4 ⎡

⎣ ⎢

⎤

⎦ ⎥1 4 2 3

Composition in Array Notation

€

βα =1 2 3 4

2 1 4 3

⎡

⎣ ⎢

⎤

⎦ ⎥

1 2 3 4

2 3 1 4

⎡

⎣ ⎢

⎤

⎦ ⎥

=1 2 3 4 ⎡

⎣ ⎢

⎤

⎦ ⎥1 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}:

€

ε =1 2 3

1 2 3

⎡

⎣ ⎢

⎤

⎦ ⎥

€

α =1 2 3

2 3 1

⎡

⎣ ⎢

⎤

⎦ ⎥

€

α 2 =1 2 3

3 1 2

⎡

⎣ ⎢

⎤

⎦ ⎥

€

β =1 2 3

1 3 2

⎡

⎣ ⎢

⎤

⎦ ⎥

€

αβ =1 2 3

2 1 3

⎡

⎣ ⎢

⎤

⎦ ⎥

€

α 2β =1 2 3

3 2 1

⎡

⎣ ⎢

⎤

⎦ ⎥

Is S3 a group?

Composition of functions is always associative.

Identity is ε. Since permutations are one to one and

onto, there exist inverses (which are also permutations.

Therefore, S3 is group.

Computations in S3

€

α 3 =1 2 3

2 3 1

⎡

⎣ ⎢

⎤

⎦ ⎥

1 2 3

3 1 2

⎡

⎣ ⎢

⎤

⎦ ⎥=

€

1 2 3

1 2 3

⎡

⎣ ⎢

⎤

⎦ ⎥= ε

€

β 2 =1 2 3

1 3 2

⎡

⎣ ⎢

⎤

⎦ ⎥ 1 2 3

1 3 2

⎡

⎣ ⎢

⎤

⎦ ⎥=

€

1 2 3

1 2 3

⎡

⎣ ⎢

⎤

⎦ ⎥= ε

€

βα =1 2 3

1 3 2

⎡

⎣ ⎢

⎤

⎦ ⎥

1 2 3

2 3 1

⎡

⎣ ⎢

⎤

⎦ ⎥=

€

1 2 3

3 2 1

⎡

⎣ ⎢

⎤

⎦ ⎥= α 2β

Simplified computations in S3

αβαβ αβααβ ααβαβ αβαβ εαββ αβ α

Double the exponent of α when switching with β.

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

23

4

€

R0 =1 2 3 4

1 2 3 4

⎡

⎣ ⎢

⎤

⎦ ⎥

€

R90 =1 2 3 4

2 3 4 1

⎡

⎣ ⎢

⎤

⎦ ⎥

€

R180 =1 2 3 4

3 4 1 2

⎡

⎣ ⎢

⎤

⎦ ⎥

€

R270 =1 2 3 4

4 1 2 3

⎡

⎣ ⎢

⎤

⎦ ⎥

€

H =1 2 3 4

2 1 4 3

⎡

⎣ ⎢

⎤

⎦ ⎥

€

V =1 2 3 4

4 3 2 1

⎡

⎣ ⎢

⎤

⎦ ⎥

€

D =1 2 3 4

1 4 3 2

⎡

⎣ ⎢

⎤

⎦ ⎥

€

′ D =1 2 3 4

3 2 1 4

⎡

⎣ ⎢

⎤

⎦ ⎥ D4 ≤ S4

Why do we care? Every group turns out to be a

permutation group on some set!

(To be proved later).

Cycle Notation

€

α =1 2 3 4

2 3 1 4

⎡

⎣ ⎢

⎤

⎦ ⎥

€

β =1 2 3 4

2 1 4 3

⎡

⎣ ⎢

⎤

⎦ ⎥

€

α =(1 2 3)

€

β =(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 αβ = (1 2 3)(1 2)(3 4)

= (1 3 4)(2)

= (1 3 4) βα = (1 2)(3 4)(1 2 3)

= (1)(2 4 3)

= (2 4 3)

Compostion in cycle notation αβ = (1 2 3)(1 2)(3 4)

= (1

βα = (1 2)(3 4)(1 2 3)

1 1 2 3