 Theorem 6.6: Let [G;] be a group and
let a and b be elements of G. Then
 (1)ac=bc, implies that a=b(right
cancellation property)。
 (2)ca=cb, implies that a=b。(left
cancellation property)
 S={a1,…,an}, al*aial*aj(ij)，
 Thus there can be no repeats in any row
or column
 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.
 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! 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.
2  n 
 1
  

 (1)  (2)   (n ) 
 1
2

n 
 i
i

i

1
2
n
  

  
  (1)  ( 2)   ( n )    (i1 )  (i2 )   (in ) 
1
identity permutation e  
1
2

2

n

n
σ(1) σ(2) σ(n)

σ1  
2
n 
 1
2
 n 
 1
1

σ   1
1
1
σ
(1)
σ
(2)

σ
(n)


inverse permutation of 
2  n 
 1
σ  
,
σ(1)σ(2)  σ(n)
2  n 
 1
τ 
,
τ(1)τ(2)  τ(n)
2  n  1
2  n 
1
σ  τ  
  

σ (1) σ (2)  σ (n)  τ (1) τ (2)  τ (n) 
τ (2)  τ (n)   1
2  n 
 τ (1)
 
  

σ (τ (1)) σ (τ (2))  σ (τ (n))  τ (1) τ (2)  τ (n) 
1
2 
n 

 

σ (τ (1)) σ (τ (2))  σ (τ (n)) 
 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 i2 id 1 id id 1 in 
  

i1 id 1  in 
 i2 i3  id
 =(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.
1 2 3 4 5 6 7 8 
σ  
  (1 3)(3 4)(2 6)(5 8)(8 7)
3 6 4 1 8 2 5 7
 (1 4)(3 1)(2 6)(5 7)(8 5)
 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.
 NEXT Cyclc groups,

Subgroups
 Exercise:P357 15,20,
 P195 8,9, 12,15,21