Chapter 5: Permutation groups;
De…nition:
A Permutation of a set A is a function from A to A that is both one-to-one
and onto.
A Permutation group of a set A is a set of Permutations of A that forms a
group under function composition.
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Example :
A={1,2,3}
1
2
3
1
2
3
1 = 1
2 = 3
3 =
3
2
1
2
1
3
2
2
1
1
4
2
2
3
1
=
3
3
1
2
2
1
3
3
5
=
1
2
2
3
3
1
6
=
Permutation Group of A = { 1 , 2 , 3 , 4 , 5 , 6 }
Solution:
1
2
3
1
2
3
= 12 21 33
1
2 = 1
3
2
3
1
2
1 2 (1) = 2,
1 2 (2) = 1,
1 2 (3) = 3;
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Example 1:
Symmetric group:
Sn ! A={1,2,. . . . . . ..n}
Sn is a Permutation Group of A.
S3 ! A={1,2,3}
S3 is a Permutation Group of A.
S4 ! A={1,2,. . . 4}
S4 is a Permutation Group of A.
Let A={1,2,3}
S3 = f 1 ; 2 ; 3 ; 4 ; 5 ; 6 g
the order of S3 = jS3 j = 3!
jS3 j=3!=3x2x1=6
jS4 j=4!=4 x3x2x1=24
jS5 j=5!=5x4x3x2x1=120
jSn j=n(n-1)(n-2). . .
jSj=n!
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Cycle notation:
= 12 21 34 46 55 63 = (1 2) (3 4 6) (5)
#
#
#
2-cycle 3-cycle 1-cycle
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Remarks:
1) (a1 ; a2 ; :::; an ) is called a cycle of length n or n-cycle.
1 2 3 4 5 6
=
= (4 6) (1) (2) (3) (5)
1 2 3 6 5 4
1
2) Let , can be express a Permutation in disjoint cycles.
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Example:
Let =(1 3) (2 7) (4 5 6) (8)
=(1 2 3 7) (6 4 8) (5)
Solution:
=(1 3) (2 7) (4 5 6) (8) (1 2 3 7) (6 4 8) (5)
=(1 7 3 2) (4 8) (5 6)
=(5 6) (8 4) (7 3 2 1)
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Theorem 5.1 :
Every Permutation of a …nite set can be written as a cycle of as a product
of disjoint cycles.
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Theorem 5.2 :
If the pair of cycles =( a1 , a2 ,. . . , an ) , =(b1 , b2 ,. . . bn )
have no entries in common
=
.
Proof:
let S={ a1 , a2 ,. . . ,am , b1 , b2 ,. . . bn , c1 , c2 ,. . . ck }
where the c’s are the number of S let …xed by both and
1)
(ai )= ( (ai ))
= (ai)
.:
(ai ) = ai+1
(ai ) = ( (ai ))
= (ai+1 ) = ai+1
(bi ) = ( (bi ))
2)
= (bi+1 )= bi+1
(bi ) = ( (bi ))
= (bi+1 )= bi+1
(ci ) = ( (ci ))= (ci )=ci
3)
(ci ) = ( (ci ))= (ci )=ci
x2 S =)
(x) =
(x)
=)
=
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Theorem 5.3:
The order of permutation of a …nite set written in disjoint cycle from is the
least common multiple of the lengths of the cycles.
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Example:
= (1 7 3 2) (4 8) (5 6)
L.C.M [4,2,2] =) L.C.M [22 ,2,2]= 22 =4
.: j
j= 4
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Example:
L.C.M [100,80,50]=)L.C.M (22 52 )(22 7 3)( 52 2)
L.C.M=22 52 7 3
2
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Note:
A permutation of the form (a,b) called transposition.
=)(ab) = (ba)
(ab) 1 = (ab)
jabj = 2
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Theorem 5.4:(product of 2-cycles)
Every permutation in Sn , n > 1, is a product of 2-cycles.
Every permutation can be written in the form: (a1 a2 ::::ak ) (b1 b2 ::::br ) (c1 c2 ::::::cs )
(a1 ak )(a1 ak 1 )..(a1 a2 )(b1 bt )(b1 bt 1 )..(b1 b2 )
...(c1 cs )
(c1 cs 1 )...(c1 c2 )
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Lemma:
if = 1 2 :::: r where the 0 s are 2-cycles, then r is even.
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Theorem 5.5:
If a permutationcan be expressed as a product of an even (odd) number of
2-cycles, then every decomposition of into a product of 2-cycles must have an
even (odd) number of 2-cycles. In symbols, if:
= 1 2 :::: r and
=
::::
1 2
s
where the 0 s and the 0 s are 2-cycles, then r and s are both even or both
odd.
Proof:
Let = 1 2 :::: r and
= 1 2 :::: s
1
= 1 2 :::: r 1 1 2 1 :::: s 1
= 1 2 :::: r 1 2 :::: s
r + s is even number.=) r; s even or r; s odd.
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De…nition:
A permutation that can be expressed as a product of an even or odd number
of 2-cycles is called an even or odd permutation.
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Theorem 5.6:
The set of even permutations in Sn forms a subgroup of Sn :
Proof:
Let H the set of even permutation.
Let 1 ; 2 2 H
, r is even number.
1 = 1 2 ::: r
, s is even number.
2 = 1 2 ::: s
=) 1 2
=) 1 2 = 1 2 ::: r 1 2 ::: s
=) 1 2 is even permutation.
3
since r + s is an even number.
=) H is clouser.
2) since is even permutation =) 2 H:
3) Let 2 H
since 2 Sn
1
=) 9
2 Sn
r
r
1
=)
= ( 1 2 ::: s )( r
1 2 ::: r )
=) s + r is even number.
since s is even =) r is even.
1
=)
is an even permutation.
1
=)
2 H =) H Sn :
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