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104A Sp09 lecture 5

# 104A Sp09 lecture 5 - 3 Associative Law A(BC =(AB)C Example...

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Symmetry and Group Theory Reading: MT, Chapter 4 Vincent, Programmes 1-4

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inversion: i inversion center or center of symmetry (x,y,z) (-x, -y, -z)
improper rotation: S n rotation about an axis by an angle of 2 π /n, followed by reflection through a perpendicular plane ( C n , σ h symmetry elements are not necessary for S n to exist)

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Group Theory Definition: A group is a collection of elements interrelated by a set of operations, which obey the following rules. Integers under the operation of addition is a specific example of a group. 1) Closure The combination (“product”) of any two elements must yield another member of the group: AB = C (the sum of two integers is an integer) 2) Identity Element Must have identity element E : E X = X E = X (the identity element for integers is zero)

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Unformatted text preview: 3) Associative Law : A(BC) = (AB)C Example: 1 + (2 + 3) = (1 + 2) + 3 4) Reciprocal Elements, S , Exist : X S = S X = E (the reciprocal of every positive integer is its negative) Key Point: Symmetry operations associated with an object form groups, so may be described and systematized using group theory. The Five Regular Polyhedra or Platonic Solids Symmetry operations: E , 8 C 3 , 6 C 2 , 6 C 4 , 3 C 2 (= C 4 2 ), i , 6 S 4 , 8 S 6 , 3 σ h , 6 d group order: 48 Symmetry operations: E , 8 C 3 , 3 C 2 , 6 S 4 , 6 d group order: 24 Tetrahedron ( T d ) Octahedron ( O h ) Icosahedron ( I h ) symmetry operations: E , 12 C 5 , 12 C 5 2 , 20 C 3 , 15 C 2 , i , 12 S 10 , 12 S 10 3 , 20 S 6 , 15 σ group order: 120...
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