Group - Group Lecture 14 Discrete Mathematic Structures Group • Group axioms – Association – Identity – Inverse property • Example –

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Unformatted text preview: Group Lecture 14 Discrete Mathematic Structures 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 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. 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 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 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, & ) 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 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 , 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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This note was uploaded on 03/31/2010 for the course SE C0229 taught by Professor Tao during the Spring '08 term at Nanjing University.

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Group - Group Lecture 14 Discrete Mathematic Structures Group • Group axioms – Association – Identity – Inverse property • Example –

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