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.

(i1 i2 …ik)=(i1 i2)(i2 i3)…(ik-2 ik-1)(ik-1 ik)

 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 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) |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 0<m<n, ame. We say that the order of a is infinite if an e for any positive integer n. Example:group[{1,-1,i.-i};], i2=-1,i3=-i, i4=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 am=e for mZ iff n|m. Example: Let the order of the element a of a group G be n. Then the order of ar is n/d, where d=(r,n) is maximum common factor of r and n. Proof: (ar)n/d=e, Let p be the order of ar. p|n/d, n/d|p p=n/d

2. Cyclic groups Definition 14: The group G is called a cyclic group if there exists gG 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.-i};],1=i0,-1=i2,-i=i3, i and –i are generators of G. [Z;+]

Example：Let the order of group G be n Example：Let the order of group G be n. If there exists gG such that g is of order n，then G is a cyclic group, and G is generated by g. Proof:

Theorem 6.15: Let [G; *] be a cyclic group, and let g be a generator of G. Then the following results hold. (1)If the order of g is infinite, then [G;*] [Z;+] (2)If the order of g is n, then [G;*][ Zn;] Proof:(1)G={gk|kZ}, :GZ, (gk)=k (2)G={e,g,g2,gn-1}, :GZn, (gk)=[k]

6.4 Subgroups, Normal subgroups and Quotient groups Definition 15: A subgroup of a group [G; *] is a nonempty subset H of G such that * is a group operation on H. Example : [Z;+] is a subgroup of the group [R; +]. G and {e} are called trivial subgroups of G, other subgroups are called proper subgroups of G.

Theorem 6. 16: Let [G;·] be a group, and H be a nonempty subset of G Theorem 6.16: Let [G;·] be a group, and H be a nonempty subset of G. Then H is a subgroup of G, iff (1) for any x, y H, x·yH; and (2) for any xH, x-1 H. Proof: If H is a subgroup of G, then (1) and (2) hold. (1) and (2) hold eH Associative Law inverse

Theorem 6. 17: Let [G;·] be a group, and H be a nonempty subset of G Theorem 6.17: Let [G;·] be a group, and H be a nonempty subset of G. Then H is a subgroup of G, iff a·b-1H for a,b H. Example: Let [H1;·] and [H2;·] be subgroups of the group [G;·]，Then [H1∩H2;·] is also a subgroup of [G;·] [H1∪H2;·] ? Example:Let G ={ (x; y)| x,yR with x 0} , and consider the binary operation ● introduced by (x, y) ● (z,w) = (xz, xw + y) for (x, y), (z, w) G. Let H ={(1, y)| yR}. Is H a subgroup of G? Why?

6.4.2 Coset Let [H;] is a subgroup of the group [G;]. We define a relation R on G, so that aRb iff for ab-1H for a,bG. The relation is called congruence relation on the subgroup [H;]. We denoted by ab(mod H)。 Theorem 6.18 ：Congruence relation on the subgroup [H;] of the group G is an equivalence relation

[a]={x|xG, and xa(mod H)}={x|xG, and xa-1H} Let h=xa-1. Then x=ha，Thus [a]={ha|hH} Ha={ha|hH} is called right coset of the subgroup H aH={ah|hH} is called left coset of the subgroup H Let [H;] be a subgroup of the group [G;], and aG. Then (1)bHa iff ba-1H (2)baH iff a-1bH

Definition 16: Let H be a subgroup of a group G, and let aG Definition 16: Let H be a subgroup of a group G, and let aG. We define the left coset of H in G containing g,written gH, by gH ={g*h| h H}. Similarity we define the right coset of H in G containing g,written Hg, by Hg ={h*g| h H}.

[E;+] E+0. E+1

NEXT : Normal subgroups and Quotient groups Exercise:P357 22—26 P362 11,12 1. Let G be a group. Suppose that a, and bG, ab=ba. If the order of a is n, and the order of b is m. Prove: (1)The order of ab is mn if (n,m)=1 (2)The order of ab is LCM(m,n) if (n,m)1 and (a)∩(b)=, where LCM(m,n) is lease common multiple of m and n.