mat301h1 - MAT301H1: Groups and Symmetries Groups and...

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MAT301H1: Groups and Symmetries Groups and Symmetries Definition: Symmetry A symmetry of a shape is a rigid motion that takes vertices to vertices, edges to edges. Note: A rigid motion preserves angles and distances. Definition: Group A group G , ∗ is a set G and an operation such that G is closed under and that: 1. There exists e such that e x = x = x e for all x G (existence of identity e ). 2. For every x G there exists x 1 such that x x 1 = e = x 1 x (existence of inverses). 3. For all x , y ,z G , x ∗ y z = x y ∗ z (associativity). Examples D 3 is the group of symmetries of a regular 3-gon. D 3 ={ e ,r ,r 2, s ,sr ,sr 2 } , D 3 , °  with r = 60° rotation clockwise and s = reflection about y-axis D n is the group of symmetries of a regular n -gon. Claim The identity of G , ∗ is unique. Proof: Assume the e 1 and e 2 are identities of G , ∗ . e 1 = e 1 e 2 = e 2 e 1 = e 2 . So there is only one identity. Claim Given x G , x 1 is unique. Proof: Let y = x 1 and z = x 1 but y z . Now y = y e = y ∗ x z = y x ∗ z = e z = z , so y = z . Contradiction! So x 1 is unique given x . Definition: Commute If x y = y x then x and y are said to commute. Definition: Abelian If all elements in G , ∗ commute, then G , ∗ is said to be Abelian. Definition: Order (Group) The order of a group G , denoted G , is the number of elements in G . Examples Q 8 ∣= 8 , V 4 ∣= 4 , D 3 ∣= 6 , D 4 ∣= 8 , ∣ℤ∣=∞ , ∣ℚ∣=∞ . Definition: Order (Element) The order of an element x G , written x , is the smallest positive integer n such that x n = e . Examples In D 3 , r r r = r 3 = e so r ∣= 3 . e ∣= 1 . 1 of 17
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MAT301H1: Groups and Symmetries In V 4 ={ e ,a ,b ,a b } , e ∣= 1 , a a = e ⇒∣ a ∣= 2 , b b = e ⇒∣ b ∣= 2 , a b ∣= 2 . In Q 8 ={ 1, 1, i , i , j , j ,k , k } , 1 ∣= 1 , ∣− 1 ∣= 2 , i ∣=∣ j ∣=∣ k ∣= 4 . Definition: Subgroup H is a subgroup of G iff H is a subset of G which is a group under the same operation as G . Example In D 4 , { e ,r ,r 2, r 3 } is a subgroup since it is closed under × ( r n × r m = r n m ), has inverses ( r 1 = r 3 , r 2 1 = r 2 ), and e ∈{ e ,r ,r 2, r 3 } . Definition: Proper Subgroup H is a proper subgroup of G iff H is a subgroup of G and H G , H ≠{ e } . Note Does G always have a proper subgroup? No. { e ,r ,r 2 } has no proper subgroups. Definition: Set of Generators S is a set of generators of a group G iff every g G can be expressed as: multiplication: g = s 1 m 1 × s 2 m 2 ×⋯× s k m k addition: g = m 1 s 1 m 2 s 2 ⋯ m k s k where m i ∈ℤ and s i S (repetitions of s i 's are allowed). Any such combinations of
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This note was uploaded on 03/26/2011 for the course MAT 301 taught by Professor Gideonmaschler during the Fall '10 term at University of Toronto.

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mat301h1 - MAT301H1: Groups and Symmetries Groups and...

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