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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 㱨 S 㱺 ( a b ) 㱨 S Associativity a , b , c 㱨 S 㱺 ( a b ) c = a ( b c ) Identity 㱽 e 㱨 S s.t. 㱼 a 㱨 S a e = e a = a Inverse 㱼 a 㱨 S 㱽 a1 㱨 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 㱨 R 5 < x < 5},+) (R, ) (Z n , +) N.B. ({x 㱨 R 5 < x < 5},+) is not closed, so it doesn’t 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 㱺 b = c b a = c a 㱺 b = c Proof a b = a c 㱻 a1 ( a b ) = a1 ( a c ) 㱻 ( a1 a ) b = ( a1 a ) c 㱻 e b = e c 㱻 b = c Unique Inverse Theorem Every element in a group has an unique inverse....
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 Fall '09
 Group Theory, Lagrange Theorem, F

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