12345678 036418252=(13)(34)(26)(58)(87) =(14)(31)(26)(57)(85)
( 1 4)(3 1)(2 6)(5 7)(8 5 ) ( 1 3)(3 4)(2 6)(5 8)(8 7 ) 3 6 4 1 8 2 5 7 1 2 3 4 5 6 7 8 σ = = =
Theorem 6.12: If a permutation of S, can be written as a product of an even number of transpositions, then it can never be written as a product of an odd number of transpositions, and conversely. Definition 12: A permutation of sn is called even it can be written as a product of an even number of transpositions, and a permutation of s is called odd if it can never be written as a product of an odd number of transpositions
Theorem 6.12: If a permutation of Sn can be written as a product of an even number of transpositions, then it can never be written as a product of an odd number of transpositions, and conversely. Definition 12 ： A permutation of Sn is called even it can be written as a product of an even number of transpositions, and a permutation of Sn is called odd if it can never be written as a product of an odd number of transpositions
〔i1i2…i1)=(i1i2)(2i3)(i12i-1)(i11 112· k-1
(i1 i2 …ik )=(i1 i2 )(i2 i3 )…(ik-2 ik-1 )(ik-1 ik ) k-1
Even permutation Odd Even permutation Even permutation Odd Odd permutation Odd permutation Even
• Even permutation Odd Even permutation Even permutation Odd Odd permutation Odd permutation Even
Even permutation odd permutation Even permutation Even permutation Odd permutation Odd permutation Odd permutation Even permutation S=OUA O.∩A= Ango is a groupo
• Even permutation odd permutation Even permutation Even permutation Odd permutation Odd permutation Odd permutation Even permutation Sn= On∪An On∩An = [An ;•] is a group