This preview shows pages 1–13. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Group Theory E d i t h L a w 2 7 . 0 3 . 2 0 0 7 Puzzle Group Theory in the Bedroom Reference: ScientiFc American, 93(5)395 Puzzle Group Theory in the Bedroom Reference: ScientiFc American, 93(5)395 What is a Group? A Familiar Group To solve the equation 4 + x = 204 + (4+x) = 4 + 20 Closure (4+4) + x = 16 Associativity 0 + x = 16 Inverse x = 16 Identity What makes this calculation possible are the abstract properties of integers under addition. Reference: Group Theory Lecture by Steven Rudich, 2000 Group An ordered pair ( S , ) where S is a set and is a binary operation on S . Closure a , b S ( a b ) S Associativity a , b , c S ( a b ) c = a ( b c ) Identity e S s.t. a S a e = e a = a Inverse a S a1 S s.t. a a1 = a1 a = e ( Z ,+) is a group Closure The sum of two integers is an integer Associativity ( a + b ) + c = a + ( b + c ) Identity For every integer a , a + 0 = 0 + a = a Inverse For every integer a , a + ( a ) = ( a ) + a = 0 Group or Not Closure Associativity Identity Inverse (Z, +) (Z{0}, ) ({x R 5 < x < 5},+) (R, ) (Z n , +) N.B. ({x R 5 < x < 5},+) is not closed, so it doesnt make sense to talk about associativity when some of the results of addition can be undeFned. Cayley Table Finite Groups can be represented by a Cayley Table. + 1 2 3 1 2 3 1 1 2 3 2 2 3 1 3 3 1 2 (Z 4 ,+) Abstraction Unique Identity Theorem A group has at most one identity element. Proof Suppose e and f are both identities of ( S , ), then f = e f = e . Cancellation Theorem Theorem The left and right cancellation laws hold. a b = a c b = c b a = c a b = c Proof a b = a c a1 ( a b ) = a1 ( a c ) ( a1 a ) b = ( a1 a ) c e b = e c b = c Unique Inverse Theorem Every element in a group has an unique inverse....
View
Full
Document
 Fall '09

Click to edit the document details