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Group - Group Lecture 14 Discrete Mathematic Structures...

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Group Lecture 14 Discrete Mathematic Structures
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Group Group axioms Association Identity Inverse property Example Addition group on integers (Z,+) All one-to-one functions on {1,2,3}, plus composition of function: S 3
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Inverse Property of a System For any element in a system, there may or may not be its inverse. However, “for any element x in S , x has its inverse” is a property of the system as a whole. For a system for which the inverse property holds, each element has its particular inverse.
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Semigroup and Group A group is a semigroup Association A group is a monoid Identity Negative exponential Denotation of inverse of element a : a -1 Expansion of exponential: a -k =(a -1 ) k (k is positive integer) Abelian group: commutative group
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An Example of Abelian Group Let G be the set of all nonzero real numbers, let a * b = ab /2, then ( G ,*) is a Abelian group. Verifying that all requirements as described as definition are satisfied: “*” is a closed binary operation on G . Associability : ( a * b )* c = a *( b * c )=( abc )/4 Identity : a *2=2* a for all a in G Inverse : examine the equation a * x = 2, it is easy to see that for all a in G , a –1 =4/ a Commutability : obviously, a * b = b * a
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Example: Let (S,° ) be group, u is a special element of S, define binary operation *: a*b = a° u -1 ° b. then (S, *) is a group G Associability G (a*b)*c = a*(b*c) = a3 u -1 o b° u -1 G c Identity G G ° °x, x*u = x° u -1 ± u =x, G u*x = u° u -1 ± x = x Inverse G G ° °x, x*(u° x -1 ± u)=x3 u -1 ¨ (u‚ x -1 ± u) = u x -1 is the inverse if x in group (S, ° )
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Some properties: Each element in group has only one inverse G be a group, a,b,c be elements of G, then ab=ac implies that b=c ba=ca implies that b=c G be a group, a,b be elements of G, then (a -1 ) -1 = a (ab) -1 = b -1 a -1
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Group Equation and Its Solution Group equations: aSx = b and ySa = b , where a , b are constants Solutions of the group equations: aSx = b aS ( a -1 G b ) = b ySa = b ( bS a -1 ) G a = b The group equation has unique solution: Assuming that aSx 1 = b = aSx 2 , multiply the two sides of the equation from the left with a -1 , x 1 = a -1 G b = x 2 ,
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Second Definition of Group ( G ,e ) is an algebraic system, if association holds for it, and the two equation aSx = b and ySa =b have unique solutions each, then ( G ) is a group.
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