Permutation Groups Part 1 Definition n A permutation

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Permutation Groups Part 1

Permutation Groups Part 1

Definition n A permutation of a set A is a function from A to

Definition n A permutation of a set A is a function from A to A that is both one to one and onto.

Array notation n n Let A = {1, 2, 3, 4} Here are two

Array notation n n Let A = {1, 2, 3, 4} Here are two permutations of A:

Composition in Array Notation 1

Composition in Array Notation 1

Composition in Array Notation 1 4

Composition in Array Notation 1 4

Composition in Array Notation 1 4 2

Composition in Array Notation 1 4 2

Composition in Array Notation 1 4 2 3

Composition in Array Notation 1 4 2 3

Composition in Array Notation 1 4 2 3

Composition in Array Notation 1 4 2 3

Definition n A permutation group of a set A is a set of permutations

Definition n A permutation group of a set A is a set of permutations of A that forms a group under function composition.

Example n n The set of all permutations on {1, 2, 3} is called

Example n n The set of all permutations on {1, 2, 3} is called the symmetric group on three letters, denoted S 3 There are 6 permutations possible:

S 3 n The permutations of {1, 2, 3}:

S 3 n The permutations of {1, 2, 3}:

Is S 3 a group? n n Composition of functions is always associative. Identity

Is S 3 a group? n n Composition of functions is always associative. Identity is . Since permutations are one to one and onto, there exist inverses (which are also permutations. Therefore, S 3 is group.

Computations in S 3

Computations in S 3

Simplified computations in S 3 Double the exponent of when switching with . You

Simplified computations in S 3 Double the exponent of when switching with . You can simplify any expression in S 3! n

Symmetric groups, Sn n Let A = {1, 2, … n}. The symmetric group

Symmetric groups, Sn n 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 S 3 is group. |Sn| = n!

Symmetries of a square, D 4 3 2 4 1 D 4 ≤ S

Symmetries of a square, D 4 3 2 4 1 D 4 ≤ S 4

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

Why do we care? n Every group turns out to be a permutation group on some set! (To be proved later).

Cycle Notation

Cycle Notation

Disjoint cycles n n Two permutations are disjoint if the sets of elements moved

Disjoint cycles n n 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 n n n Let permutation π be given. Let a

Algorithm for disjoint cycles n n n 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 n n = (1 2 3)(1 2)(3 4) = (1

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

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

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