Section 4.1 Finite Permutation Groups

Permutation of a Set Let A be the set { 1, 2, …, n }. A permutation on A is a function f : A  A that is both one-to-one and onto. The set of all permutations on A is denoted by S n A permutation is represented by a matrix :

Examples of Permutation Let A be the set { 1, 2, 3, 4, 5 } f and g are elements of S 5

Product of Permutations The product of f and g is the composition function f 。 g

Cycles : A special kind of Permutation An element f of S n is a cycle (r-cycle) if there exists such that Cycles will be written simply as ( i 1, i 2,..., i r ) Example : = (1, 3, 4)

Product of Cycles If (1, 3, 2, 4) and (1, 2, 6, 7) are two cycles in S 7 we have (1, 3, 2, 4) (1, 2, 6, 7) = (1, 4) (2, 6, 7, 3) Note that (1, 4) (2, 6, 7, 3) is a product of disjoint cycles

Permutations and cycles Every permutation can be written as a product of disjoint cycles. For example We have 1  3  2  8  1 4  6  4 5  7  9  5 We can easily verify that f = (1, 3, 2, 8)(4, 6)(5, 7, 9)

Transpositions : A special kind of cycles A 2-cycle such as (3, 7) is called a transposition Every cycle can be written as a product of transposition : ( i 1, i 2,..., i r ) = ( i 1, i r )( i 1, i r-1 )... ( i 1, i 3 )( i 1, i 2 ) For example, (1, 3, 2, 4) = (1, 4)(1, 2)(1, 3)

Permutations and transpositions Since every permutation can be expressed as a product of (disjoint) cycles, every permutation can be expressed as a product of transpositions For example,

Product of Transposition Theorem 4.3 If a permutation f is expressed as a product of p transpositions and also a product of q transpositions, then p and q are either both even or both odd. Definition 4.4 A permutation that can be expressed as a product of an even number of transpositions is called an even permutation, and is called an odd permutation if it can be expressed a product of odd transpositions

Even, Odd Permutations Observe that (1,3,2,4)(1,7,6,2) = (1,7,6,4)(2,3) So, we can write this permutation as two different product of transpositions : (1,3,2,4)(1,7,6,2) = (1,4)(1,2)(1,3)(1,2)(1,6)(1,7) (1,7,6,4)(2,3) = (1,4)(1,6)(1,7)(2,3) However, Theorem 4.3 shows that they must be both even or both odd (it’s even in this case) Note that (1,2)(1,2) is an expression of the identity mapping, so identity is an even permutation

Application (7, 8) is a transposition in S 9 In fact,

Application We may as well use this table to represent the transposition (7, 8)

Application So =

Application What is the transposition representation after switching 7 and 9? ( Should be (?, ?)(7, 8) )

Application What is the transposition representation of the table? YES, it is (7, 9)(7, 8) Because : (7,9)(7,8) = (7, 8, 9)

Sam Loyd ’ s Puzzle Eight Goal : To arrange the number in order Rule : The only allowed moves are sliding numbers into the empty square Is it Possible?