1. 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.

2. 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

3. 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°)

4. 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

5. 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.

6. 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

7. 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.

11. 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.

12. =(i1,…, id):
A 2-cycle  is called transposition.
Theorem 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.

14. 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.

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

16.  Even permutation Odd
Even permutation Even permutation Odd
Odd permutation Odd permutation Even

17.  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。

18. 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

19. Exercise:P ,20,
P195 8,9, 12,15,21