permutation groups part 1. definition a permutation of a set a is a function from a to a that is...
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![Page 1: Permutation Groups Part 1. Definition A permutation of a set A is a function from A to A that is both one to one and onto](https://reader030.vdocument.in/reader030/viewer/2022032709/56649edc5503460f94bec785/html5/thumbnails/1.jpg)
Permutation Groups
Part 1
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Definition A permutation of a set A is a function
from A to A that is both one to one and onto.
![Page 3: Permutation Groups Part 1. Definition A permutation of a set A is a function from A to A that is both one to one and onto](https://reader030.vdocument.in/reader030/viewer/2022032709/56649edc5503460f94bec785/html5/thumbnails/3.jpg)
Array notation Let A = {1, 2, 3, 4} Here are two permutations of A:
€
α =1 2 3 4
2 3 1 4
⎡
⎣ ⎢
⎤
⎦ ⎥
€
β =1 2 3 4
2 1 4 3
⎡
⎣ ⎢
⎤
⎦ ⎥
€
α(2) = 3
€
α(4) = 4
€
β(4) = 3
€
β(1) = 2
€
βα(2) = β (3) = 4
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Composition in Array Notation
€
βα =1 2 3 4
2 1 4 3
⎡
⎣ ⎢
⎤
⎦ ⎥
1 2 3 4
2 3 1 4
⎡
⎣ ⎢
⎤
⎦ ⎥
=1 2 3 4 ⎡
⎣ ⎢
⎤
⎦ ⎥1
![Page 5: Permutation Groups Part 1. Definition A permutation of a set A is a function from A to A that is both one to one and onto](https://reader030.vdocument.in/reader030/viewer/2022032709/56649edc5503460f94bec785/html5/thumbnails/5.jpg)
Composition in Array Notation
€
βα =1 2 3 4
2 1 4 3
⎡
⎣ ⎢
⎤
⎦ ⎥
1 2 3 4
2 3 1 4
⎡
⎣ ⎢
⎤
⎦ ⎥
=1 2 3 4 ⎡
⎣ ⎢
⎤
⎦ ⎥1 4
![Page 6: Permutation Groups Part 1. Definition A permutation of a set A is a function from A to A that is both one to one and onto](https://reader030.vdocument.in/reader030/viewer/2022032709/56649edc5503460f94bec785/html5/thumbnails/6.jpg)
Composition in Array Notation
€
βα =1 2 3 4
2 1 4 3
⎡
⎣ ⎢
⎤
⎦ ⎥
1 2 3 4
2 3 1 4
⎡
⎣ ⎢
⎤
⎦ ⎥
=1 2 3 4 ⎡
⎣ ⎢
⎤
⎦ ⎥1 4 2
![Page 7: Permutation Groups Part 1. Definition A permutation of a set A is a function from A to A that is both one to one and onto](https://reader030.vdocument.in/reader030/viewer/2022032709/56649edc5503460f94bec785/html5/thumbnails/7.jpg)
Composition in Array Notation
€
βα =1 2 3 4
2 1 4 3
⎡
⎣ ⎢
⎤
⎦ ⎥
1 2 3 4
2 3 1 4
⎡
⎣ ⎢
⎤
⎦ ⎥
=1 2 3 4 ⎡
⎣ ⎢
⎤
⎦ ⎥1 4 2 3
![Page 8: Permutation Groups Part 1. Definition A permutation of a set A is a function from A to A that is both one to one and onto](https://reader030.vdocument.in/reader030/viewer/2022032709/56649edc5503460f94bec785/html5/thumbnails/8.jpg)
Composition in Array Notation
€
βα =1 2 3 4
2 1 4 3
⎡
⎣ ⎢
⎤
⎦ ⎥
1 2 3 4
2 3 1 4
⎡
⎣ ⎢
⎤
⎦ ⎥
=1 2 3 4 ⎡
⎣ ⎢
⎤
⎦ ⎥1 4 2 3
![Page 9: Permutation Groups Part 1. Definition A permutation of a set A is a function from A to A that is both one to one and onto](https://reader030.vdocument.in/reader030/viewer/2022032709/56649edc5503460f94bec785/html5/thumbnails/9.jpg)
Definition A permutation group of a set A is a set
of permutations of A that forms a group under function composition.
![Page 10: Permutation Groups Part 1. Definition A permutation of a set A is a function from A to A that is both one to one and onto](https://reader030.vdocument.in/reader030/viewer/2022032709/56649edc5503460f94bec785/html5/thumbnails/10.jpg)
Example The set of all permutations on {1,2,3} is
called the symmetric group on three letters, denoted S3
There are 6 permutations possible:
€
1 2 3
__ __ __
⎡
⎣ ⎢
⎤
⎦ ⎥
3 × 2 × 1 = 6
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S3
The permutations of {1,2,3}:
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ε =1 2 3
1 2 3
⎡
⎣ ⎢
⎤
⎦ ⎥
€
α =1 2 3
2 3 1
⎡
⎣ ⎢
⎤
⎦ ⎥
€
α 2 =1 2 3
3 1 2
⎡
⎣ ⎢
⎤
⎦ ⎥
€
β =1 2 3
1 3 2
⎡
⎣ ⎢
⎤
⎦ ⎥
€
αβ =1 2 3
2 1 3
⎡
⎣ ⎢
⎤
⎦ ⎥
€
α 2β =1 2 3
3 2 1
⎡
⎣ ⎢
⎤
⎦ ⎥
![Page 12: Permutation Groups Part 1. Definition A permutation of a set A is a function from A to A that is both one to one and onto](https://reader030.vdocument.in/reader030/viewer/2022032709/56649edc5503460f94bec785/html5/thumbnails/12.jpg)
Is S3 a group?
Composition of functions is always associative.
Identity is ε. Since permutations are one to one and
onto, there exist inverses (which are also permutations.
Therefore, S3 is group.
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Computations in S3
€
α 3 =1 2 3
2 3 1
⎡
⎣ ⎢
⎤
⎦ ⎥
1 2 3
3 1 2
⎡
⎣ ⎢
⎤
⎦ ⎥=
€
1 2 3
1 2 3
⎡
⎣ ⎢
⎤
⎦ ⎥= ε
€
β 2 =1 2 3
1 3 2
⎡
⎣ ⎢
⎤
⎦ ⎥ 1 2 3
1 3 2
⎡
⎣ ⎢
⎤
⎦ ⎥=
€
1 2 3
1 2 3
⎡
⎣ ⎢
⎤
⎦ ⎥= ε
€
βα =1 2 3
1 3 2
⎡
⎣ ⎢
⎤
⎦ ⎥
1 2 3
2 3 1
⎡
⎣ ⎢
⎤
⎦ ⎥=
€
1 2 3
3 2 1
⎡
⎣ ⎢
⎤
⎦ ⎥= α 2β
![Page 14: Permutation Groups Part 1. Definition A permutation of a set A is a function from A to A that is both one to one and onto](https://reader030.vdocument.in/reader030/viewer/2022032709/56649edc5503460f94bec785/html5/thumbnails/14.jpg)
Simplified computations in S3
αβαβ αβααβ ααβαβ αβαβ εαββ αβ α
Double the exponent of α when switching with β.
You can simplify any expression in S3!
![Page 15: Permutation Groups Part 1. Definition A permutation of a set A is a function from A to A that is both one to one and onto](https://reader030.vdocument.in/reader030/viewer/2022032709/56649edc5503460f94bec785/html5/thumbnails/15.jpg)
Symmetric groups, Sn
Let A = {1, 2, … n}. The symmetric group on n letters, denoted Sn, is the group of all permutations of A under composition.
Sn is a group for the same reasons that S3 is group.
|Sn| = n!
![Page 16: Permutation Groups Part 1. Definition A permutation of a set A is a function from A to A that is both one to one and onto](https://reader030.vdocument.in/reader030/viewer/2022032709/56649edc5503460f94bec785/html5/thumbnails/16.jpg)
Symmetries of a square, D4
1
23
4
€
R0 =1 2 3 4
1 2 3 4
⎡
⎣ ⎢
⎤
⎦ ⎥
€
R90 =1 2 3 4
2 3 4 1
⎡
⎣ ⎢
⎤
⎦ ⎥
€
R180 =1 2 3 4
3 4 1 2
⎡
⎣ ⎢
⎤
⎦ ⎥
€
R270 =1 2 3 4
4 1 2 3
⎡
⎣ ⎢
⎤
⎦ ⎥
€
H =1 2 3 4
2 1 4 3
⎡
⎣ ⎢
⎤
⎦ ⎥
€
V =1 2 3 4
4 3 2 1
⎡
⎣ ⎢
⎤
⎦ ⎥
€
D =1 2 3 4
1 4 3 2
⎡
⎣ ⎢
⎤
⎦ ⎥
€
′ D =1 2 3 4
3 2 1 4
⎡
⎣ ⎢
⎤
⎦ ⎥ D4 ≤ S4
![Page 17: Permutation Groups Part 1. Definition A permutation of a set A is a function from A to A that is both one to one and onto](https://reader030.vdocument.in/reader030/viewer/2022032709/56649edc5503460f94bec785/html5/thumbnails/17.jpg)
Why do we care? Every group turns out to be a
permutation group on some set!
(To be proved later).
![Page 18: Permutation Groups Part 1. Definition A permutation of a set A is a function from A to A that is both one to one and onto](https://reader030.vdocument.in/reader030/viewer/2022032709/56649edc5503460f94bec785/html5/thumbnails/18.jpg)
Cycle Notation
€
α =1 2 3 4
2 3 1 4
⎡
⎣ ⎢
⎤
⎦ ⎥
€
β =1 2 3 4
2 1 4 3
⎡
⎣ ⎢
⎤
⎦ ⎥
€
α =(1 2 3)
€
β =(1 2)(3 4)
![Page 19: Permutation Groups Part 1. Definition A permutation of a set A is a function from A to A that is both one to one and onto](https://reader030.vdocument.in/reader030/viewer/2022032709/56649edc5503460f94bec785/html5/thumbnails/19.jpg)
Disjoint cycles Two permutations are disjoint if the sets
of elements moved by the permutations are disjoint.
Every permutation can be represented as a product of disjoint cycles.
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Algorithm for disjoint cycles Let permutation π be given. Let a be the identity permutation,
represented by an empty list of cycles. while there exists n with π(n) ≠ a(n):
start a new cycle with n
let b = n
while
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Compostion in cycle notation αβ = (1 2 3)(1 2)(3 4)
= (1 3 4)(2)
= (1 3 4) βα = (1 2)(3 4)(1 2 3)
= (1)(2 4 3)
= (2 4 3)
![Page 22: Permutation Groups Part 1. Definition A permutation of a set A is a function from A to A that is both one to one and onto](https://reader030.vdocument.in/reader030/viewer/2022032709/56649edc5503460f94bec785/html5/thumbnails/22.jpg)
Compostion in cycle notation αβ = (1 2 3)(1 2)(3 4)
= (1
βα = (1 2)(3 4)(1 2 3)
1 1 2 3