Theorem 6.7: Let [G;] be a group and let a, b, and c be elements of G. Then (1)The equation ax=b has a unique solution in G. (2)The equation ya=b has a unique solution in G.

a-k=(a-1)k， ak=a*ak-1(k≥1) Let [G;] be a group. We define a0=e， a-k=(a-1)k， ak=a*ak-1(k≥1) Theorem 6.8: Let [G;] be a group and a G, m,n Z. Then (1)am*an=am+n (2)(am)n=amn a+a+…+a=ma, ma+na=(m+n)a n(ma)=(nm)a

6.3 Permutation groups and cyclic groups Example: Consider the equilateral triangle with vertices 1，2，and 3. Let l1, l2, and l3 be the angle bisectors of the corresponding angles, and let O be their point of intersection。 Counterclockwise rotation of the triangle about O through 120°,240°,360° (0°)

f2:12，23，31 f3:13，21，32 f1 :11，22，33 reflect the lines l1, l2, and l3. g1:11，23，32 g2:13，22，31 g3:12，21，33

6.3.1 Permutation groups Definition 9: A bijection from a set S to itself is called a permutation of S Lemma 6.1：Let S be a set. (1) Let f and g be two permutations of S. Then the composition of f and g is a permutation of S. (2) Let f be a permutation of S. Then the inverse of f is a permutation of S.

Theorem 6.9：Let S be a set. The set of all permutations of S, under the operation of composition of permutations, forms a group A(S). Proof: Lemma 6.1 implies that the rule of multiplication is well-defined. associative. the identity function from S to S is identity element The inverse permutation g of f is a permutation of S

Theorem 6. 10: Let S be a finite set with n elements. Then A(S) has n Theorem 6.10: Let S be a finite set with n elements. Then A(S) has n! elements. Definition 10: The group Sn is the set of permutations of the first n natural numbers. The group is called the symmetric group on n letters, is called also the permutation group.

Definition 11: Let |S|=n, and let Sn Definition 11: Let |S|=n, and let Sn.We say that  is a d-cycle if there are integers i1; i2; … ; id such that (i1) =i2, (i2) = i3, … , and (id) =i1 and  fixes every other integer, i.e.

=(i1,…, id): A 2-cycle  is called transposition. Theorem 6.11. Let  be any element of Sn. Then  may be expressed as a product of disjoint cycles. Corollary 6.1. Every permutation of Sn is a product 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.

(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

Exercise:P357 15,20, P195 8,9, 12,15,21