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
Theorem 6.13: The set of even permutations forms a group, is called the altemating group of degree n and denoted by An. The order of An is n.2( where n>1) 1An=? n=1,|An=1。 n>1。 An|=|O=n!/2
Theorem 6.13: The set of even permutations forms a group, is called the altemating group of degree n and denoted by An . The order of An is n!/2( where n>1) |An |=? n=1，|An |=1。 n>1, |An |=|On |=n!/2
6.3.2 Cyclic groups 1. Order of an element Definition 13: Let G be a group with an identity element e. We say that a is of order n if an=e, and for any (<m<n, amte. We say that the order of a is infinite if an*e for any positive integer n. Example: groupl(l, -1,i-i; x1 i4-=1 (-i)2=-1,(-i)3=i,(-i)4=1
6.3.2 Cyclic groups 1.Order of an element Definition 13: Let G be a group with an identity element e. We say that a is of order n if a n =e, and for any 0<m<n, a me. We say that the order of a is infinite if a n e for any positive integer n. Example:group[{1,-1,i.-i};], i 2=-1,i3=-i, i 4=1 (-i)2=-1, (-i)3=i, (-i)4=1
Theorem 6.14: Let a is an element of the group G, and let its order be n. Then a= e for n∈ Ziff nn. Example: Let the order of the element a of a group g be n. Then the order of ar is n/d, where d=( n) is maximum common factor of r and n Proof:(ar)/d=e, Let p be the order of ar. pn/d, n/dp p=n/d
Theorem 6.14: Let a is an element of the group G, and let its order be n. Then a m=e for mZ iff n|m. Example: Let the order of the element a of a group G be n. Then the order of a r is n/d, where d=(r,n) is maximum common factor of r and n. Proof: (a r ) n/d=e, Let p be the order of a r . p|n/d, n/d|p p=n/d
2. CV yclic groups Definition 14: The group G is called a cyclic group if there exists gEG such that h= gk for any h∈G, where k∈ Z We say that g is a generator of G We denoted by g-g Example: group{1,-1,-i};×,1=,-1=i2,=, i and -i are generators of G. Z;+
2. Cyclic groups Definition 14: The group G is called a cyclic group if there exists gG such that h=gk for any hG , where kZ.We say that g is a generator of G. We denoted by G=(g). Example:group[{1,-1,i.-i};],1=i0 ,-1=i2 ,-i=i3 , i and –i are generators of G. [Z;+]
Example: Let the order of group g be n If there exists gEG such that g is of order n,then G is a cyclic group, and G is generated by g. Proof:
Example：Let the order of group G be n. If there exists gG such that g is of order n，then G is a cyclic group, and G is generated by g. Proof:
