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

??? ??? 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- -01 01- -2 2- -0522 0522 Agreement Number F30602 This work is supported by: This work is supported by: Definition of a Group Definition of a Group The Defense Advance Research Projects • G Definition. Definition . A A group group is a set together with a is a set together with a Agency (DARPA) & Air Force Research binary operation satisfying the : G G G binary operation satisfying the • • × × → Laboratory (AFRL), Air Force Materiel Command, following axioms: following axioms: USAF Agreement Number F30602-01-2-0522. This definition took This definition took g i ( ( g g i ) ) ( ( g g i ) ) i g , g g g , , G • = = ∀ ∀ ∈ 1 2 3 1 2 3 1 2 3 The National Institute for Standards • 100s of years 1 • There exists a unique element , called the There exists a unique element , called the 100s of years and Technology (NIST) identity, such that , such that identity 1 i g g g i 1, g G = = = = ∀ ∈ to develop ! • The Mathematical Sciences Research to develop ! 1 g − • g G , there exists a unique element , , there exists a unique element , ∀ ∈ ∀ ∈ Institute (MSRI). g called the inverse called the inverse of , such that of , such that • The Institute for Scientific Interchange g g i 1 1 g 1 i g − − − − = = = = Why is it so important ? Why is it so important ? • The L-O-O-P Fund. L- L -O O- -O O- -P P Definition of a Group Definition of a Group Purpose Purpose Definition. Definition . A group A group is a set together with a is a set together with a G A Group is a A Group is a binary operation satisfying the binary operation satisfying the : G G G • • × × → following axioms: following axioms: Mathematical Tool for g i ( ( g g i ) ) ( ( g g i ) ) i g , g g g , , G Mathematical Tool for • = = ∀ ∀ ∈ 1 2 3 1 2 3 1 2 3 1 • There exists a unique element , called the There exists a unique element , called the Quantifying Quantifying identity, such that identity , such that 1 i g g g i 1, g G = = = = ∀ ∈ 1 g − , there exists a unique element , • g G , there exists a unique element , ∀ ∈ ∀ ∈ Symmetry g Symmetry called the inverse inverse of , such that of , such that called the 1 1 g g i − − 1 g − − i g = = = = 1

Example: Symmetries of the Equilateral Triangle Symmetries of the Equilateral Triangle Example: Symmetries of the Equilateral Triangle Symmetries of the Equilateral Triangle Example: Example: 1 2 2 1 ρ ρ Rotation Rotation Rotation Rotation 3 1 2 3 1 3 2 3 1 1 1 1 σ σ Reflection Reflection Reflection Reflection 2 3 2 2 3 2 3 3 We Can Multiply Symmetries We Can Multiply Symmetries We Can Multiply Symmetries We Can Multiply Symmetries 2 ρ 1 1 2 1 3 1 1 ρ ρ ρ σ σ 2 3 3 1 3 2 3 3 2 2 1 2 2 3 2 1 3 1 ∴ ∴ σ = = ∴ ∴ ρ = = Therefore, we have the relation relation 3 Therefore, we have the relation relation 2 Therefore, we have the 1 Therefore, we have the 1 ρ = σ = There is also a relation There is also a relation between the symmetries and between the symmetries and ρ σ The Group of Symmetries of the The Group of Symmetries of the Equilateral Triangle is Given by the Equilateral Triangle is Given by the 2 2 σρ σρ = ρσ σρ σρ Group P Presentation resentation Group 1 2 3 Dihedral Group Dihedral Group D ρ Relations Relations ρ Generators Generators 3 2 3 2 3 1 1 3 2 2 ( ( ) ) , : 1, 1, ρ σ ρ = = σ σ = = σ σρ ρ = ρ ρσ σ 1 σ 3 ρ 3 2 2 1 Every Relation Among Every Relation Among Symmetries is a Symmetries is a Every Symmetry is a Every Symmetry is a ρσ consequence of these consequence of these Composition of these Composition of these 2

Example: Symmetries of the Symmetries of the Example: Symmetries of the Regular Symmetries of the Regular n n- -gon gon Example: Example: Oriented Equilateral Triangle Oriented Equilateral Triangle ( ( 3 ) ) � : 1 = = ρ ρ ρ ρ = = ( ( n 2 n 1 ) ) , : 1, 1, − 3 ρ σ ρ = = σ = = ρ σ σ = σ σρ Cyclic Group of Order 3 Cyclic Group of Order 3 D Dihedral Group Dihedral Group n Example: Symmetries of Oriented Regular Symmetries of Oriented Regular n n- -gon gon Example: More Generally, a group More Generally, a group presentation presentation is of is of the following form: the following form: n ( ( ) ) : 1 ρ ρ ρ ρ = x x , , … , x : r 1, r 1, r 1 ( ( ) ) = = = = = 0 1 n 1 0 1 m 1 − − − − Relations Relations Generators Generators Cyclic Group of Order n Cyclic Group of Order n � n Free Groups Free Abelian Abelian Groups Groups Free Groups Free A group F F is free if the only relations A group A A is free is free abelian abelian if the only if the only A group is free if the only relations A group among its generators are those required relations among its generators are those among its generators are those required relations among its generators are those for F for F to be a group to be a group required for required for A A to be an to be an abelian abelian group group F ( ( x x , , … , x − : ) ) ( ( ) ) A x , x , … , x : x x x x i j , = = = = = ∀ 0 1 n 1 0 1 n 1 i j j i − x x − = 1 x x − = 1 • Allowed: Allowed: • Allowed: Allowed: 1 1 i i i i x x x x , i j • Not Allowed: Not Allowed: = = ≠ ≠ x x x x , i j • Allowed: Allowed: = = ≠ ≠ i j j i i j j i 3 x 1 • Not Allowed: Not Allowed: 3 = x 1 • Not Allowed: Not Allowed: = i i 3

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