Lecture13 - 15-251 Fall 2011 Algebraic Structures Groups Theory Il est peu de notions en mathematiques qui soient plus primitives que celle de loi

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10/11/2011 1 15-251: Fall 2011 Algebraic Structures: Groups Theory Lecture 13 (October 11, 2011) Il est peu de notions en mathematiques qui soient plus primitives que celle de loi de composition. - Nicolas Bourbaki Number Theory Naturals closed under + a+b = b+a a+0 = 0+a=a Integers closed under + a+b = b+a a+0 = 0+a=a a+(-a) = 0 Z n closed under + n a+ n b = b+ n a a+ n 0 = 0+ n a=a a+ n (-a) = 0 (a+b)+c = a+(b+c) (a+b)+c = a+(b+c) (a+ n b)+ n c = a+ n (b+ n c) A+0 =A a+0 = a a+(-a) = 0 a+ n 0 = a a+ n (-a) = 0 closed under * closed under * n closed under * (a+b)*c = a*c+b*c ditto ditto A+(-A) = 0 1/a may not exist ditto ditto Number Theory closed under + A+B = B+A closed under + a+b = b+a closed under + n a+ n b = b+ n a (a+b)+c = a+(b+c) (A+B)+C = A+(B+C) Matrices Integers Z n (a+ n b)+ n c = a+ n (b+ n c) A+0 = 0+A a+0 = 0+a a+(-a) = 0 a+ n 0 = 0+ n a a+ n (-a) = 0 closed under * closed under * n closed under * (a+b)*c = a*c+b*c ditto ditto A+(-A) = 0 1/a exists if a 0 ditto ditto Number Theory closed under + A+B = B+A closed under + a+b = b+a closed under + n a+ n b = b+ n a (a+b)+c = a+(b+c) (a+ n b)+ n c = a+ n (b+ n c) (A+B)+C = A+(B+C) Invertible Matrices Rationals Z n (n prime) Abstraction : Abstract away the inessential features of a problem =
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10/11/2011 2 Today we are going to study the abstract properties of binary operations Rotating a Square in Space Imagine we can pick up the square, rotate it in any way we want, and then put it back on the white frame In how many different ways can we put the square back on the frame? R 90 R 180 R 270 R 0 F | F F F We will now study these 8 motions, called symmetries of the square Symmetries of the Square Y SQ = { R 0 , R 90 , R 180 , R 270 , F | , F , F , F } Composition Define the operation “ ” to mean “first do one symmetry, and then do the next” For example, R 90 R 180 Question: if a,b Y SQ , does a b Y SQ ? Yes! means “first rotate 90˚ clockwise and then 180˚” = R 270 F | R 90 means “first flip horizontally and then rotate 90˚” = F R 90 R 180 R 270 R 0 F | F F F R 0 R 90 R 180 R 270 F | F F F R 0 R 90 R 180 R 270 F | F F F R 90 R 180 R 270 F | F F F R 180 R 270 R 0 R 270 R 0 R 90 R 0 R 90 R 180 F F F | F F F | F F F F F F | F F F F F | F F F F | F | F F R 0 R 0 R 0 R 0 R 180 R 90 R 270 R 180 R 270 R 90 R 270 R 90 R 180 R 90 R 270 R 180
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10/11/2011 3 How many symmetries for n-sided body? 2n R 0 , R 1 , R 2 , …, R n-1 F 0 , F 1 , F 2 , …, F n-1 R i R j = R i+j R i F j = F j-i F j R i = F j+i F i F j = R j-i Some Formalism If S is a set, S S is: the set of all (ordered) pairs of elements of S S S = { (a,b) | a S and b S } Formally, is a function from Y SQ Y SQ to Y SQ : Y SQ Y SQ Y SQ As shorthand, we write (a,b) as “a b” ” is called a binary operation on Y SQ Definition: A binary operation on a set S is a function : S S S Example: The function f:
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This note was uploaded on 11/03/2011 for the course CS 251 taught by Professor Gupta during the Spring '11 term at Carnegie Mellon.

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Lecture13 - 15-251 Fall 2011 Algebraic Structures Groups Theory Il est peu de notions en mathematiques qui soient plus primitives que celle de loi

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