??? ??? Group = Symmetry Group = Symmetry Samuel J. Lomonaco, Jr. - - PDF document

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??? ??? Group = Symmetry Group = Symmetry

Samuel J. Lomonaco, Jr.

• Dept. of Comp. Sci. & Electrical Engineering

University of Maryland Baltimore County Baltimore, MD 21250 Email: Lomonaco@UMBC.EDU WebPage: http://www.csee.umbc.edu/~lomonaco Defense Advanced Research Projects Agency (DARPA) & Defense Advanced Research Projects Agency (DARPA) & Air Force Research Laboratory, Air Force Materiel Command, USAF Air Force Research Laboratory, Air Force Materiel Command, USAF Agreement Number F30602 Agreement Number F30602-

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2-

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• The Defense Advance Research Projects

Agency (DARPA) & Air Force Research Laboratory (AFRL), Air Force Materiel Command, USAF Agreement Number F30602-01-2-0522.

• The National Institute for Standards

and Technology (NIST)

• The Mathematical Sciences Research

Institute (MSRI).

• The Institute for Scientific Interchange
• The L-O-O-P Fund.

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P

This work is supported by: This work is supported by:

Definition Definition. . A A group group is a set together with a is a set together with a binary operation satisfying the binary operation satisfying the following axioms: following axioms:

Definition of a Group Definition of a Group G :G G G

• ×
• ×

→

• There exists a unique element , called the

There exists a unique element , called the identity identity, such that , such that

• , there exists a unique element ,

, there exists a unique element , called the called the inverse inverse of , such that

• f , such that

( ) ( ) ( ) ( )

1 2 3 1 2 3 1 2 3

, , , g g g g g g g g g G = ∀ = ∀ ∈ i i i i 1 1, g g g g G = = = = ∀ ∈ i i g G ∀ ∈ ∀ ∈

1

1

g−

1 1

1 g g g g

− − − −

= = = = i i g

This definition took This definition took

• f years
• f years

to develop ! to develop !

Why is it so important ? Why is it so important ?

100s 100s A Group is a A Group is a Mathematical Tool for Mathematical Tool for

Quantifying Quantifying Symmetry Symmetry

Purpose Purpose

Definition Definition. . A A group group is a set together with a is a set together with a binary operation satisfying the binary operation satisfying the following axioms: following axioms:

Definition of a Group Definition of a Group G :G G G

• ×
• ×

→

• There exists a unique element , called the

There exists a unique element , called the identity identity, such that , such that

• , there exists a unique element ,

, there exists a unique element , called the called the inverse inverse of , such that

• f , such that

( ) ( ) ( ) ( )

1 2 3 1 2 3 1 2 3

, , , g g g g g g g g g G = ∀ = ∀ ∈ i i i i 1 1, g g g g G = = = = ∀ ∈ i i g G ∀ ∈ ∀ ∈

1

1

g−

1 1

1 g g g g

− − − −

= = = = i i g

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2

Example: Example: Symmetries of the Equilateral Triangle

Symmetries of the Equilateral Triangle

1 2 3 ρ

Rotation Rotation

1 2 3 1 2 3

Reflection Reflection

σ 1 2 3

Example: Example: Symmetries of the Equilateral Triangle

Symmetries of the Equilateral Triangle

1 2 3 1 2 3 ρ

Rotation Rotation

1 1 2 2 3 3

Reflection Reflection

σ

We Can Multiply Symmetries We Can Multiply Symmetries

1 2 3 1 2 3 ρ 1 2 3 1 3 2 ρ ρ

3

1 ρ ∴ = ∴ =

2

ρ

Therefore, we have the Therefore, we have the relation relation

3

1 ρ =

We Can Multiply Symmetries We Can Multiply Symmetries

1 2 3 1 3 2 σ 1 3 2 σ

2

1 σ ∴ = ∴ =

Therefore, we have the Therefore, we have the relation relation

2

1 σ =

There is also a There is also a relation relation between the symmetries and between the symmetries and

1 2 3 1 2 3 ρ 1 2 3 ρ ρ σ 2 3 1 1 3 2 σ σ ρ

2

σρ σρ ρσ

2

σρ σρ ρσ =

The Group of Symmetries of the The Group of Symmetries of the Equilateral Triangle is Given by the Equilateral Triangle is Given by the Group Group P Presentation resentation

( ) ( )

3 2 2

, : 1, 1, ρ σ ρ σ σ σ σρ ρ ρ ρσ = = = = =

Generators Generators Relations Relations Every Symmetry is a Every Symmetry is a Composition of these Composition of these Every Relation Among Every Relation Among Symmetries is a Symmetries is a consequence of these consequence of these

Dihedral Group Dihedral Group

3

D

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Example: Example: Symmetries of the

Symmetries of the Oriented Equilateral Triangle Oriented Equilateral Triangle

( ) ( )

3 3

: 1 ρ ρ ρ ρ = = = =

• Cyclic Group of Order 3

Cyclic Group of Order 3

Example: Example: Symmetries of the Regular

Symmetries of the Regular n n-

• gon

gon

( ) ( )

2 1

, : 1, 1,

n n

ρ σ ρ σ ρ σ σ σ σρ

−

= = = = =

Dihedral Group Dihedral Group

n

D

Example: Example: Symmetries of Oriented Regular

Symmetries of Oriented Regular n n-

• gon

gon

( ) ( )

: 1

n

ρ ρ ρ ρ =

Cyclic Group of Order Cyclic Group of Order n n

n

• More Generally, a

More Generally, a group group presentation presentation is of is of the following form: the following form:

( ) ( )

1 1 1 1

, , , : 1, 1, 1

n m

x x x r r r

− − − −

= = = = = …

Generators Generators Relations Relations Free Groups Free Groups

( ) ( )

1 1

, , , :

n

F x x x − = …

A group A group F F is free if the only relations is free if the only relations among its generators are those required among its generators are those required for for F F to be a group to be a group

• Allowed:

Allowed:

1

1

i i

x x − =

• Not Allowed:

Not Allowed:

,

i j j i

x x x x i j = ≠ = ≠

• Not Allowed:

Not Allowed:

3

1

i

x =

Free Free Abelian Abelian Groups Groups

( ) ( )

1 1

, , , : ,

n i j j i

A x x x x x x x i j

−

= = = = ∀ …

A group A group A A is free is free abelian abelian if the only if the only relations among its generators are those relations among its generators are those required for required for A A to be an to be an abelian abelian group group

• Allowed:

Allowed:

• Allowed:

Allowed:

• Not Allowed:

Not Allowed:

1

1

i i

x x − =

3

1

i

x = ,

i j j i

x x x x i j = ≠ = ≠

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